Section 1.1

  1. 11. If y=y1=x2, then y(x)=2x3 and y(x)=6x4, so x2y+5xy+4y=x2(6x4)+5x(2x3)+4(x2)=6x210x2+4x2=0. If y=y2=x2ln x, then y(x)=x32x3ln x and y(x)=5x4+6x4ln x, so x2y+5xy+4y=x2(5x4+6x4ln x)+5x(x32x3ln x)+4(x2ln x)=0.

  2. 13. r=23

  3. 14. r=±12

  4. 15. r=2, 1

  5. 16. r=16(3±57)

  6. 17. C=2

  7. 18. C=3

  8. 19. C=6

  9. 20. C=11

  10. 21. C=7

  11. 22. C=1

  12. 23. C=56

  13. 24. C=17

  14. 25. C=π/4

  15. 26. C=π

  16. 27. y=x+y

  17. 28. y=2y/x

  18. 29. y=x/(1y)

  19. 31. y=(yx)/(y+x)

  20. 32. dP/dt=kP

  21. 33. dv/dt=kv2

  22. 35. dN/dt=k(PN)

  23. 37. y1 or y=x

  24. 39. y=x2

  25. 41. y=12ex

  26. 42. y=cosx or y=sinx

  27. 43. (b) The identically zero function x(0)0

  28. 44. (a) The graphs (figure below) of typical solutions with k=12 suggest that (for each) the value x(t) increases without bound as t increases.

    (b) The graphs (figure below) of typical solutions with k=12 suggest that now the value x(t) approaches 0 as t increases without bound.

  29. 45. P(t)=100/(50t); P=100 when t=49, and P=1000 when t=49.9. Thus it appears that P(t) grows without bound as t approaches 50.

  30. 46. v(t)=50/(5+2t); v=1 when t=22.5, and v=110 when t=247.5. Thus it appears that v(t) approaches 0 as t increases without bound.

  31. 47. (a) C=10.1; (b) No such C, but the constant function y(x)0 satisfies the conditions y=y2 and y(0)=0.

Section 1.2

  1. 1. y(x)=x2+x+3

  2. 2. y(x)=13(x2)3+1

  3. 3. y(x)=13(2x3/216)

  4. 4. y(x)=1/x+6

  5. 5. y(x)=2x+25

  6. 6. y(x)=13[(x2+9)3/2125]

  7. 7. y(x)=10tan1 x

  8. 8. y(x)=12sin 2x+1

  9. 9. y(x)=sin1 x

  10. 10. y(x)=(x+1)ex+2

  11. 11. x(t)=25t2+10t+20

  12. 12. x(t)=10t215t+5

  13. 13. x(t)=12t3+5t

  14. 14. x(t)=13t3+12t27t+4

  15. 15. x(t)=13(t+3)437t26

  16. 16. x(t)=43(t+4)3/25t293

  17. 17. x(t)=12[(t+1)1+t1]

  18. 19. x(t)={5tif 0t5,10t12t2252if 5t10.

  19. 20. x(t)={12t2if 0t5,5t252if 5t10.

  20. 21. x(t)={12t2if 0t5,10t12t225if 5t10.

  21. 22. x(t)={56t2if 0t3,5t152if 3t7,16(5t2+100t290)if 7t10.

  22. 23. v(t)=(9.8)t+49, so the ball reaches its maximum height (v=0) after t=5 seconds. Its maximum height then is y(5)=122.5(m).

  23. 24. v(5)=160 ft/s

  24. 25. The car stops when t2.78 (s), so the distance traveled before stopping is approximately x(2.78)38.58 (m).

  25. 26. (a) y530 m (b) t20.41 s (c) t20.61 s

  26. 27. y0178.57 (m)

  27. 28. v(4.77)192.64 ft/s

  28. 29. After 10 seconds the car has traveled 200 ft and is traveling at 70 ft/s.

  29. 30. a=22 ft/s2; it skids for 4 seconds.

  30. 31. v0=1030 (m/s), about 197.18 km/h

  31. 32. 60 m

  32. 33. 201063.25 (ft/s)

  33. 34. 460.8 ft

  34. 36. About 13.6 ft

  35. 37. 25 (mi)

  36. 38. 1:10 pm

  37. 39. 6 mph

  38. 40. 2.4 mi

  39. 41. 5443181.33 (ft/s)

  40. 42. 25 mi

  41. 43. Time: 6.12245×109 s194 years;

    Distance: 1.8367×1017 m19.4 light-years

  42. 44. About 54 mi/h

Section 1.3

  1. 1.

  2. 2.

  3. 3.

  4. 4.

  5. 5.

  6. 6.

  7. 7.

  8. 8.

  9. 9.

  10. 10.

  11. 11. A unique solution exists in some neighborhood of x=1.

  12. 12. A unique solution exists in some neighborhood of x=1.

  13. 13. A unique solution exists in some neighborhood of x=0.

  14. 14. Existence but not uniqueness is guaranteed in some neighborhood of x=0.

  15. 15. Neither existence nor uniqueness is guaranteed in any neighborhood of x=2.

  16. 16. A unique solution exists in some neighborhood of x=2.

  17. 17. A unique solution exists in some neighborhood of x=0.

  18. 18. Neither existence nor uniqueness is guaranteed.

  19. 19. A unique solution exists in some neighborhood of x=0.

  20. 20. A unique solution exists in some neighborhood of x=0.

  21. 21. Your figure should suggest that y(4)3; an exact solution of the differential equation gives y(4)=3+e43.0183.

  22. 22. y(4)3

  23. 23. Your figure should suggest that y(2)1; the actual value is closer to 1.004.

  24. 24. y(2)1.5

  25. 25. Your figure should suggest that the limiting velocity is about 20 ft/sec (quite survivable) and that the time required to reach 19 ft/sec is a little less than 2 seconds. An exact solution gives v(t)=19 when t=58ln 201.8723.

  26. 26. A figure suggests that there are 40 deer after about 60 months; a more accurate value is t61.61. The limiting population is 75 deer.

  27. 27. The initial value problem y=2y, y(0)=b has no solution if b<0; a unique solution if b>0; infinitely many solutions if b=0.

  28. 28. The initial value problem xy=y, y(a)=b has a unique solution if a0; infinitely many solutions if a=b=0; no solution if a=0 but b0.

  29. 29. The initial value problem y=3y2/3, y(a)=b always has infinitely many solutions defined for all x. However, if b0 then it has a unique solution near x=a.

  30. 30. The initial value problem y=1y2, y(a)=b has a unique solution if |b|<1; no solution if |b|>1, and infinitely many solutions (defined for all x) if b=±1.

  31. 31. The initial value problem y=1y2, y(a)=b has a unique solution if |b|<1; no solution if |b|>1, and infinitely many solutions (defined for all x) if b=±1.

  32. 32. The initial value problem y=4xy, y(a)=b has infinitely many solutions (defined for all x) if b0; no solutions if b<0. However, if b>0 then it has a unique solution near x=a.

  33. 33. The initial value problem x2y+y2=0, y(a)=b has a unique solution with initial point (a, b) if a0, no solution if a=0 but b0, infinitely many solutions if a=b=0.

  34. 34. (a) If y(1)=1.2 then y(1)0.48. If y(1)=0.8 then y(1)2.48. (b) If y(3)=3.01 then y(3)1.0343.

    If y(3)=2.99 then y(3)7.0343.

  35. 35. (a) If y(3)=0.2 then y(2)2.019. If y(3)=+0.2 then y(2)2.022. In either case, y(2)2.02. (b) If y(3)0.5 then y(2)2.017. If y(3)+0.5 then y(2)2.024. In either case, y(2)2.02.

Section 1.4

  1. 1. y(x)=Cexp(x2)

  2. 2. y(x)=1/(x2+C)

  3. 3. y(x)=Cexp(cos x)

  4. 4. y(x)=C(1+x)4

  5. 5. y(x)=sin (C+x)

  6. 6. y(x)=(x3/2+C)2

  7. 7. y(x)=(2x4/3+C)3/2

  8. 8. y(x)=sin1(x2+C)

  9. 9. y(x)=C(1+x)/(1x)

  10. 10. y(x)=(1+x)/[1+C(1+x)]1

  11. 11. y(x)=(Cx2)1/2

  12. 12. y2+1=Cex2

  13. 13. ln(y4+1)=C+4 sinx

  14. 14. 3y+2y3/2=3x+2x3/2+C

  15. 15. 1/(3y3)2/y=1/x+ln|x|+C

  16. 16. y(x)=sec1(C1+x2)

  17. 17. ln|1+y|=x+12x2+C

  18. 18. y(x)=tan(C1xx)

  19. 19. y(x)=2 exp(ex)

  20. 20. y(x)=tan(x3+π/4)

  21. 21. y2=1+x216

  22. 22. y(x)=3 exp(x4x)

  23. 23. y(x)=12(1+e2x2)

  24. 24. y(x)=π2sin x

  25. 25. y(x)=x exp(x21)

  26. 26. y(x)=1/(1x2x3)

  27. 27. y=ln(3e2x2)

  28. 28. y(x)=tan1(x1)

  29. 29. (a) General solution y(x)=1/(xC); (b) The singular solution y(x)0. (c) In the following figure we see that there is a unique solution through every point of the xy-plane.

  30. 30. General solution y(x)=(xC)2; singular solution y(x)0. (a) No solution if b<0; (b) Infinitely many solutions (for all x) if b0; (c) Two solutions near (a, b) if b>0.

  31. 31. Separation of variables gives the same general solution y=(xC)2 as in Problem 30 , but the restriction that y=2y0 implies that only the right halves of the parabolas qualify as solution curves. In the figure below we see that through the point (a, b) there passes (a) No solution curve if b<0, (b) a unique solution curve if b>0, (c) Infinitely many solution curves if b=0.

  32. 32. General solution y(x)=±sec(xC); singular solutions y(x)±1.

    (a) No solution if |b|<1; (b) A unique solution if |b|>1; (c) Infinitely many solutions if b=±1.

  33. 33. About 51840 persons

  34. 34. t 3.87 hr

  35. 35. About 14735 years

  36. 36. Age about 686 years

  37. 37. $21103:48

  38. 38. $44.52

  39. 39. 2585 mg

  40. 40. About 35 years

  41. 41. About 4.86×109 years ago

  42. 42. About 1.25 billion years

  43. 43. After a total of about 63 min have elapsed

  44. 44. About 2.41 minutes

  45. 45. (a) 0.495 m; (b (8.32×107)I0; (c) 3.29 m

  46. 46. (a) About 9.60 inches; (b) About 18,200 ft

  47. 47. After about 46 days

  48. 48. About 6 billion years

  49. 49. After about 66 min 40 s

  50. 50. (a) A(t)=10·32t/15; (b) About 20.80 pu; (c) About 15.72 years

  51. 51. (a) A(t)=15·(23)t/5; (b) approximately 7.84 su; (c) After about 33.4 months

  52. 52. About 120 thousand years ago

  53. 53. About 74 thousand years ago

  54. 54. 3 hours

  55. 55. 972 s

  56. 56. At time t=2048/15621.31 (in hours)

  57. 58. 1:20 p.m.

  58. 59. (a) y(t)=(87t)2/3; (b) at 1:08:34 p.m.; (c) r=1607120.15 (in.)

  59. 60. About 6 min 3 sec

  60. 61. Approximately 14 min 29 s

  61. 62. The tank is empty about 14 seconds after 2:00 p.m.

  62. 63. (a) 1:53:34 p.m.; (b) r0.04442 ft 0.53 in.

  63. 64. r=17203 ft, about 135 in.

  64. 65. At approximately 10:29 a.m.

Section 1.5

  1. 1. y(x)=2(1ex)

  2. 2. y(x)=(3x+C)e2x

  3. 3. y(x)=e3x(x2+C)

  4. 4. y(x)=(x+C)ex2

  5. 5. y(x)=x+4x2

  6. 6. y(x)=x2+32/x5

  7. 7. y(x)=5x1/2+Cx1/2

  8. 8. y(x)=3x+Cx1/3

  9. 9. y(x)=x(7+lnx)

  10. 10. y(x)=3x3+Cx3/2

  11. 11. y(x)0

  12. 12. y(x)=14x556x3

  13. 13. y(x)=(ex+ex)/2

  14. 14. y(x)=x3 ln x+10x3

  15. 15. y(x)=[15 exp(x2)]/2

  16. 16. y(x)=1+esinx

  17. 17. y(x)=(1+sinx)/(1+x)

  18. 18. y(x)=x2(sinx+C)

  19. 19. y(x)=12sinx+Ccscx

  20. 20. y(x)=1+exp(x+12x2)

  21. 21. y(x)=x3 sinx

  22. 22. y(x)=(x3+5)ex2

  23. 23. y(x)=x3(2+Ce2x)

  24. 24. y(x)=13[1+16(x2+4)3/2]

  25. 25. y(x)=[exp(32x2)][3(x2+1)3/22]

  26. 26. x(y)=1/2y2+C/y4

  27. 27. x(y)=ey(C+12y2)

  28. 28. x(y)=12[y+(1+y2)(tan1y+C)]

  29. 30. y(x)=x1/21xt1/2cost dt

  30. 29. y(x)=[exp(x2)][C+12π erf(x)]

  31. 32. (a) y(x)=sinxcosx; (b) y(x)=Cex+sinxcosx; (c) y(x)=2ex+sinxcosx

  32. 33. After about 7 min 41 s

  33. 34. About 22.2 days

  34. 35. About 5.5452 years

  35. 36. (a) x(t)=(60t)(60t)3/3600; (b) About 23.09 lb

  36. 37. 393:75 lb

  37. 38. (a) x(t)=50et/20; (b) y(t)=150et/40100et/20; (c) 56.25 lb

  38. 39. (b) ymax=100e136.79 (gal)

  39. 41. (b) Approximately $1,308,283

  40. 43. 50.0529, 28.0265, 6.0000, 16.0265, 38.0529

  41. 44. 3.99982, 4.00005, 4.00027, 4.00050, 4.00073

  42. 45. x(t)=20(1et/10); x=10 after t=10 ln 26.93 months.

  43. 46. x(t)=20101(101102et/10+cos t+10 sin t); x=10 after t=6.47 months.

Section 1.6

  1. 1. x22xyy2=C

  2. 2. y2=x2(lnx+C)

  3. 3. y(x)=x(C+ln |x|)2

  4. 4. 2 tan1(y/x)ln(y2/x2+1)=2 ln x+C

  5. 5. ln |xy|=C+xy1

  6. 6. 2y lny=x+Cy

  7. 7. y3=3x3(C+ln|x|)

  8. 8. y=x ln(Clnx)

  9. 9. y(x)=x/(Cln|x|)

  10. 10. x2+xy2=Cx6

  11. 11. y=C(x2+y2)

  12. 12. 4x2+y2=x2(ln x+C)2

  13. 13. y+x2+y2=Cx2

  14. 14. xx2+y2=C

  15. 15. x2(2xy+y2)=C

  16. 16. x=2x+y+12 ln(1+x+y+1)+C

  17. 17. y(x)=4x+2 tan(2x+C)

  18. 18. y=ln(x+y+1)+C

  19. 19. y2=x/(2+Cx5)

  20. 20. y3=3+Ce3x2

  21. 21. y2=1/(Ce2x1)

  22. 22. y3=7x/(7Cx7+15)

  23. 23. y(x)=(x+Cx2)3

  24. 24. y2=e2x/(C+ln x)

  25. 25. 2x3y3=31+x4+C

  26. 26. y3=ex(x+C)

  27. 27. y(x)=(x4+Cx)1/3

  28. 28. y=ln(Cx2+x2e2x)

  29. 29. sin2y=4x2+Cx

  30. 30. x22xeye2y=C

  31. 31. x2+3xy+y2=C

  32. 32. 2x2xy+3y2=C

  33. 33. x3+2xy2+2y3=C

  34. 34. x3+x2y2+y4=C

  35. 35. 3x4+4y3+12y ln x=C

  36. 36. x+exy+y2=C

  37. 37. sin x+x ln y+ey=C

  38. 38. x2+2x tan1y+ln(1+y2)=C

  39. 39. 5x3y3+5xy4+y5=C

  40. 40. exsin y+xtan y=C

  41. 41. x2y1+y2x3+2y1/2=C

  42. 42. xy2/3+x3/2y=C

  43. 43. y(x)=Ax2+B

  44. 44. x(y)=Ay2+B

  45. 45. y(x)=A cos 2x+Bsin 2x

  46. 46. y(x)=x2+A ln x+B

  47. 47. y(x)=Aln |x+B|

  48. 48. y(x)=lnx+Ax2+B

  49. 49. y(x)=±(A+Bex)1/2

  50. 50. y(x)=ln|sec(x+A)|12x2+B

  51. 51. x(y)=13(y3+Ay+B)

  52. 52. Ay2(Ax+B)2=1

  53. 53. y(x)=Atan(Ax+B)

  54. 54. Ay2(Bx)=1

  55. 58. y=exp(x2+C/x2)

  56. 59. x22xyy22x6y=C

  57. 60. (x+3y+3)5=C(yx5)

  58. 61. x=tan(xy)+sec(xy)+C

  59. 64. y(x)=x+ex2[C+12π erf(x)]1

  60. 65. y(x)=x+(Cx)1

  61. 69. Approximately 3.68 mi

Chapter 1 Review Problems

  1. 1. Linear: y(x)=x3(C+ln x)

  2. 2. Separable: y(x)=x/(3Cxxln x)

  3. 3. Homogeneous: y(x)=x/(Cln x)

  4. 4. Exact: x2y3+excos y=C

  5. 5. Separable: y(x)=Cexp(x3x2)

  6. 6. Separable: y(x)=x/(1+Cx+2xln x)

  7. 7. Linear: y(x)=x2(C+ln x)

  8. 8. Homogeneous: y(x)=3Cx/(Cx3)=3x/(1+Kx3)

  9. 9. Bernoulli: y(x)=(x2+Cx1)2

  10. 10. Separable: y(x)=tan(C+x+13x3)

  11. 11. Homogeneous: y(x)=x/(C3 ln x)

  12. 12. Exact: 3x2y3+2xy4=C

  13. 13. Separable: y(x)=1/(C+2x2x5)

  14. 14. Homogeneous: y2=x2/(C+2 ln x)

  15. 15. Linear: y(x)=(x3+C)e3x

  16. 16. Substitution: v=yx; solution: yx1=Ce2x(yx+1)

  17. 17. Exact: ex+ey+exy=C

  18. 18. Homogeneous: y2=Cx2(x2y2)

  19. 19. Separable: y(x)=x2/(x5+Cx2+1)

  20. 20. Linear: y(x)=2x3/2+Cx3

  21. 21. Linear: y(x)=[C+ln(x1)]/(x+1)

  22. 22. Bernoulli: y(x)=(2x4+Cx2)3

  23. 23. Exact: xey+y sin x=C

  24. 24. Separable: y(x)=x1/2/(6x2+Cx1/2+2)

  25. 25. Linear: y(x)=(x+1)2(x3+3x2+3x+C)=x+1+K(x+1)2

  26. 26. Exact: 3x3/2y4/35x6/5y3/2=C

  27. 27. Bernoulli: y(x)=x1(C+ln x)1/3

  28. 28. Linear: y(x)=x1(C+e2x)

  29. 29. Linear: y(x)=(x2+x+C)(2x+1)1/2

  30. 30. Substitution: v=x+y; solution: x=2(x+y)1/22 ln[1+(x+y)1/2]+C

  31. 31. Separable and linear

  32. 32. Separable and Bernoulli

  33. 33. Exact and homogeneous

  34. 34. Exact and homogeneous

  35. 35. Separable and linear

  36. 36. Separable and Bernoulli

Chapter 2

Section 2.1

  1. 1. x(t)=22et

  2. 2. x(t)=101+9e10t

  3. 3. x(t)=2+e2t2e2t

  4. 4. x(t)=3(1e12t)2(1+e12t)

  5. 5. x(t)=4083e15t

  6. 6. x(t)=102+3e15t

  7. 7. x(t)=77114e28t

  8. 8. x(t)=221174e91t

  9. 9. 484

  10. 10. 20 weeks

  11. 11. (b) P(t)=(12t+10)2

  12. 12. P(t)=24020t

  13. 13. P(t)=18030t

  14. 14. P(t)=P01+kP0t

  15. 16. About 27.69 months

  16. 17. About 44.22 months

  17. 19. About 24.41 months

  18. 20. About 42.12 months

  19. 21. 2001+e6/5153.7 million

  20. 22. About 34.66 days

  21. 23. (a) limtx(t)=200 grams (b) 54ln 31.37 seconds

  22. 24. About 9.24 days

  23. 25. (a) M=100 and k=0.0002; (b) In the year 2075

  24. 26. 50 ln 9815.89 months

  25. 27. (a) 100 ln9558.78 months; (b) 100 ln 269.31 months.

  26. 28. (a) The alligators eventually die out.(b) Doomsday occurs after about 9 years 2 months.

  27. 29. (a) P(140)127.008 million (b) About 210.544 million; (c) In 2000 we get P196.169, whereas the actual 2000 population was about 281.422 million.

  28. 31. a0.3915;2.15×106 cells

  29. 37. k0.0000668717, M338.027

  30. 38. k0.000146679, M208.250

  31. 39. P(t)=P0exp(kt+b2πsin 2 πt); the colored curve in the figure below shows the graph with P0=100, k=0.03, and b=0.06. It oscillates about the black curve which represents natural growth with P0=100 and k=0.03. We see that the two agree at the end of each full year.

Section 2.2

  1. 1. Unstable critical point: x=4;

    x(t)=4+(x04)et

  2. 2. Stable critical point: x=3;

    x(t)=3+(x03)et

  3. 3. Stable critical point: x=0; unstable critical point: x=4;

    x(t) = 4x0x0+(4x0)e4t

  4. 4. Stable critical point: x=3; unstable critical point: x=0;

    x(t) = 3x0x0+(3x0)e3t

  5. 5. Stable critical point: x=2; unstable critical point: x=2;

    x(t) = 2[x0+2+(x02)e4t]x0+2+(x02)e4t

  6. 6. Stable critical point: x=3; unstable critical point: x=3;

    x(t) = 3[x03+(x0+3)e6t]x03+(x0+3)e4t

  7. 7. Semi-stable (see Problem 18 ) critical point: x=2;

    x(t) = (2t1)x04ttx02t1

  8. 8. Semi-stable critical point: x=3;

    x(t) = (3t+1)x09ttx03t+1

  9. 9. Stable critical point: x=1; unstable critical point: x=4;

    x(t) = 4(1x0)+(x04)e3t(1x0)+(x04)e3t

  10. 10. Stable critical point: x=5; unstable critical point: x=2;

    x(t) =2(5x0)+5(x02)e3t(5x0)+(x02)e3t

  11. 11. Unstable critical point: x=1;

    1(x(t)2)2=1(x02)22t

  12. 12. Stable critical point: x=2;

    1(2x(t))2=1(2x0)2+2t

For each of Problems 13 through 18 we show a plot of slope field and typical solution curves. The equilibrium solutions of the given differential equation are labeled, and the stability or instability of each should be clear from the picture.

  1. 13.

  2. 14.

  3. 15.

  4. 16.

  5. 17.

  6. 18.

  7. 19. There are two critical points if h<212, one critical point if h=212, and no critical points if h>212. The bifurcation diagram is the parabola (c5)2=2510h in the hc-plane.

  8. 20. There are two critical points if s<116, one critical point if s=116, and no critical points if s>116. The bifurcation diagram is the parabola (2c5)2=25(116s) in the sc-plane.

Section 2.3

  1. 1. Approximately 31.5 s

  2. 3. 400/(ln 2)577 ft

  3. 5. 400 ln 7778 ft

  4. 7. (a) 100 ft/sec; (b) about 23 sec and 1403 ft to reach 90 ft/sec

  5. 8. (a) 100 ft/sec; (b) about 14.7 sec and 830 ft to reach 90 ft/sec

  6. 9. 50 ft/s

  7. 10. About 5 min 47 s

  8. 11. Time of fall: about 12.5 s

  9. 12. Approximately 648 ft

  10. 19. Approximately 30.46 ft/s; exactly 40 ft/s

  11. 20. Approximately 277.26 ft

  12. 22. Approximately 20.67 ft/s; about 484.57 s

  13. 23. Approximately 259.304 s

  14. 24. (a) About 0:88 cm; (b) about 2:91 km

  15. 25. (b) About 1.389 km/sec; (c) rmax=100R/195.26R

  16. 26. Yes

  17. 28. (b) After about 812 minutes it hits the surface at about 4.116 km/sec.

  18. 29. About 51.427 km

  19. 30. Approximately 11.11 km/sec (as compared with the earth’s escape velocity of about 11.18 km/sec).

Section 2.4

In Problems 1 through 10 we round off the indicated values to 3 decimal places.

  1. 1. Approximate values 1.125 and 1.181; true value 1.213

  2. 2. Approximate values 1.125 and 1.244; true value 1.359

  3. 3. Approximate values 2.125 and 2.221; true value 2.297

  4. 4. Approximate values 0.625 and 0.681; true value 0.713

  5. 5. Approximate values 0.938 and 0.889; true value 0.851

  6. 6. Approximate values 1.750 and 1.627; true value 1.558

  7. 7. Approximate values 2.859 and 2.737; true value 2.647

  8. 8. Approximate values 0.445 and 0.420; true value 0.405

  9. 9. Approximate values 1.267 and 1.278; true value 1.287

  10. 10. Approximate values 1.125 and 1.231; true value 1.333

Problems 11 through 24 call for tables of values that would occupy too much space for inclusion here. In Problems 11 through 16 we give first the final x-value, next the corresponding approximate x-values obtained with step sizes h=0.01 and h=0.005, and then the final true y-value. (All y-values rounded off accurate to 4 decimal places.)

  1. 11. 1.0, 0.7048, 0.7115, 0.7183

  2. 12. 1.0, 2.9864, 2.9931, 3.0000

  3. 13. 2.0, 4.8890, 4.8940, 4.8990

  4. 14. 2.0, 3.2031, 3.2304, 3.2589

  5. 15. 3.0, 3.4422, 3.4433, 3.4444

  6. 16. 3.0, 8.8440, 8.8445, 8.8451

In Problems 17 through 24 we give first the final x-value and then the corresponding approximate y-values obtained with step sizes h=0.1, h=0.02, h=0.004, and h=0.0008 respectively. (All y-values rounded off accurate to 4 decimal places.)

  1. 17. 1.0, 0.2925, 0.3379, 0.3477, 0.3497

  2. 18. 2.0, 1.6680, 1.6771, 1.6790, 1.6794

  3. 19. 2.0, 6.1831, 6.3653, 6.4022, 6.4096

  4. 20. 2.0, 1.3792, 1.2843, 1.2649, 1.2610

  5. 21. 2.0, 2.8508, 2.8681, 2.8716, 2.8723

  6. 22. 2.0, 6.9879, 7.2601, 7.3154, 7.3264

  7. 23. 1.0, 1.2262, 1.2300, 1.2306, 1.2307

  8. 24. 1.0, 0.9585, 0.9918, 0.9984, 0.9997

  9. 25. With both step sizes h=0.01 and h=0.005, the approximate velocity after 1 second is 16.0 ft/sec (80% of the limiting velocity of 20 ft/sec); after 2 seconds it is 19.2 ft/sec (96% of the limiting velocity).

  10. 26. With both step sizes h=1 and h=0.5, the approximate population after 5 years is 49 deer (65% of the limiting population of 75 deer); after 10 years it is 66 deer (88% of the limiting population).

  11. 27. With successive step sizes h=0.1, 0.01, 0.001, the first four approximations to y(2) we obtain are 0.7772, 0.9777, 1.0017, and 1.0042. It therefore seems likely that y(2)1.00.

  12. 28. With successive step sizes h=0.1, 0.01, 0.001, the first four approximations to y(2) we obtain are 1.2900, 1.4435, 1.4613, and 1.4631. It therefore seems likely that y(2)1.46.

  13. 29.

    x h=0.15
    y
    h=0.03
    y
    h=0.006
    y
    1.0 1.0000 1.0000 1.0000
    0.7 1.0472 1.0512 1.0521
    0.4 1.1213 1.1358 1.1390
    0.1 1.2826 1.3612 1.3835
    0.2 0.8900 1.4711 0.8210
    0.5 0.7460 1.2808 0.7192
  14. 30.

    x h=0.1
    y
    h=0.01
    y
    1.8 2.8200 4.3308
    1.9 3.9393 7.9425
    2.0 5.8521 28.3926
  15. 31.

    x h=0.1
    y
    h=0.01
    y
    0.7 4.3460 6.4643
    0.8 5.8670 11.8425
    0.9 8.3349 39.5010

Section 2.5

  1. 1.

    x Improved Euler y Actual y
    0.1 1.8100 1.8097
    0.2 1.6381 1.6375
    0.3 1.4824 1.4816
    0.4 1.3416 1.3406
    0.5 1.2142 1.2131

Note: In Problems 2 through 10, we give the value of x, the corresponding improved Euler value of y, and the true value of y.

  1. 2. 0.5, 1.3514, 1.3191

  2. 3. 0.5, 2.2949, 2.2974

  3. 4. 0.5, 0.7142, 0.7131

  4. 5. 0.5, 0.8526, 0.8513

  5. 6. 0.5, 1.5575, 1.5576

  6. 7. 0.5, 2.6405, 2.6475

  7. 8. 0.5, 0.4053, 0.4055

  8. 9. 0.5, 1.2873, 1.2874

  9. 10. 0.5, 1.3309, 1.3333

In Problems 11 through 16 we give the final value of x, the corresponding values of y with h=0.01 and with h=0.005, and the true value of y

  1. 11. 1.0, 0.71824, 0.71827, 0.71828

  2. 12. 1.0, 2.99995, 2.99999, 3.00000

  3. 13. 2.0, 4.89901, 4.89899, 4.89898

  4. 14. 2.0, 3.25847, 3.25878, 3.25889

  5. 15. 3.0, 3.44445, 3.44445, 3.44444

  6. 16. 3.0, 8.84511, 8.84509, 8.84509

In Problems 17 through 24 we give the final value of x and the corresponding values of y for h=0.1, 0.02, 0.004, and 0:0008.

  1. 17. 1.0, 0.35183, 0.35030, 0.35023, 0.35023

  2. 18. 2.0, 1.68043, 1.67949, 1.67946, 1.67946

  3. 19. 2.0, 6.40834, 6.41134, 6.41147, 6.41147

  4. 20. 2.0, 1.26092, 1.26003, 1.25999, 1.25999

  5. 21. 2.0, 2.87204, 2.87245, 2.87247, 2.87247

  6. 22. 2.0, 7.31578, 7.32841, 7.32916, 7.32920

  7. 23. 1.0, 1.22967, 1.23069, 1.23073, 1.23073

  8. 24. 1.0, 1.00006, 1.00000, 1.00000, 1.00000

  9. 25. With both step sizes h=0.01 and h=0.005 the approximate velocity after 1 second is 15.962 ft/sec (80% of the limiting velocity of 20 ft/sec); after 2 seconds it is 19.185 ft/sec (96% of the limiting velocity).

  10. 26. With both step sizes h=1 and h=0.5 the approximate population after 5 years is 49.391 deer (65% of the limiting population of 75 deer); after 10 years it is 66.113 deer (88% of the limiting population).

  11. 27. With successive step sizes h=0.1, 0.01, 0.001, the first three approximations to y(2) we obtain are 1.0109, 1.0045, and 1.0045. It therefore seems likely that y(2)1.0045.

  12. 28. With successive step sizes h=0.1, 0.01, 0.001, the first four approximations to y(2) we obtain are 1.4662, 1.4634, 1.4633, and 1.4633. It therefore seems likely that y(2)1.4633.

  13. 29. Impact speed approximately 43.22 m/s

  14. 30. Impact speed approximately 43.48 m/s

Section 2.6

  1. 1. y(0.25)1.55762; y(0.25)=1.55760. y(0.5)1.21309; y(0.5)=1.21306. Solution: y=2ex

In Problems 2 through 10 we give the approximation to y(0.5), its true value, and the solution.

  1. 2. 1.35867, 1.35914; y=12e2x

  2. 3. 2.29740, 2.29744; y=2ex1

  3. 4. 0.71309, 0.71306; y=2ex+x1

  4. 5. 0.85130, 0.85128; y=ex+x+2

  5. 6. 1.55759, 1.55760; u=2 exp(x2)

  6. 7. 2.64745, 2.64749; y=3 exp(x3)

  7. 8. 0.40547, 0.40547; y=ln(x+1)

  8. 9. 1.28743, 1.28743; y=tan14(x+π)

  9. 10. 1.33337, 1.33333; y=(1x2)1

  10. 11. Solution: y(x)=2ex.

    x h=0.2
    y
    h=0.1
    y
    Exact y
    0.0 1.00000 1.00000 1.00000
    0.2 0.77860 0.77860 0.77860
    0.4 0.50818 0.50818 0.50818
    0.6 0.17789 0.17788 0.17788
    0.8 0.22552 0.22554 0.22554
    1.0 0.71825 0.71828 0.71828

In Problems 12 through 16 we give the final value of x, the corresponding Runge-Kutta approximations with h=0.2 and with h=0.1, the exact value of y, and the solution.

  1. 12. 1.0, 2.99996, 3.00000, 3.00000; y=1+2/(2x)

  2. 13. 2.0, 4.89900, 4.89898, 4.89898; y=8+x4

  3. 14. 2.0, 3.25795, 3.25882, 3.25889; y=1/(1ln x)

  4. 15. 3.0, 3.44445, 3.44444, 3.44444; y=x+4x2

  5. 16. 3.0, 8.84515, 8.84509, 8.84509; y=(x637)1/3

In Problems 17 through 24 we give the final value of x and the corresponding values of y with h=0.2, 0.1, 0.05, and 0.025.

  1. 17. 1.0, 0.350258, 0.350234, 0.350232, 0.350232

  2. 18. 2.0, 1.679513, 1.679461, 1.679459, 1.679459

  3. 19. 2.0, 6.411464, 6.411474, 6.411474, 6.411474

  4. 20. 2.0, 1.259990, 1.259992, 1.259993, 1.259993

  5. 21. 2.0, 2.872467, 2.872468, 2.872468, 2.872468

  6. 22. 2.0, 7.326761, 7.328452, 7.328971, 7.329134

  7. 23. 1.0, 1.230735, 1.230731, 1.230731, 1.230731

  8. 24. 1.0, 1.000000, 1.000000, 1.000000, 1.000000

  9. 25. With both step sizes h=0.1 and h=0.05, the approximate velocity after 1 second is 15.962 ft/sec (80% of the limiting velocity of 20 ft/sec); after 2 seconds it is 19.185 ft/sec (96% of the limiting velocity).

  10. 26. With both step sizes h=6 and h=3, the approximate population after 5 years is 49.3915 deer (65% of the limiting population of 75 deer); after 10 years it is 66.1136 deer (88% of the limiting population).

  11. 27. With successive step sizes h=1, 0.1, 0.01, the first four approximations to y(2) we obtain are 1.05722, 1.00447, 1.00445and 1.00445. Thus it seems likely that y(2)1.00445 accurate to 5 decimal places.

  12. 28. With successive step sizes h=1, 0.1, 0.01, the first four approximations to y(2) we obtain are 1.48990, 1.46332, 1.46331, and 1.46331. Thus it seems likely that y(2)1.4633 accurate to 5 decimal places.

  13. 29. Time aloft: approximately 9.41 seconds

  14. 30. Time aloft: approximately 9.41 seconds

Chapter 3

Section 3.1

  1. 1. x=3, y=2

  2. 2. x=5, y=3

  3. 3. x=4, y=3

  4. 4. x=5, y=4

  5. 5. Inconsistent—no solution

  6. 6. Inconsistent—no solution

  7. 7. x=10+4t, y=t (infinitely many solutions)

  8. 8. x=4+2t, y=t (infinitely many solutions)

  9. 9. x=4, y=1, z=3

  10. 10. x=3, y=1, z=2

  11. 11. x=1, y=3, z=4

  12. 12. x=1, y=3, z=5

  13. 13. x=y=z=0

  14. 14. x=5, y=3, z=4

  15. 15. Inconsistent—no solution

  16. 16. Inconsistent—no solution

  17. 17. Inconsistent—no solution

  18. 18. Inconsistent—no solution

  19. 19. x=8+3t, y=3+2t, z=t (infinitely many solutions)

  20. 20. x=5t, y=5+t, z=t (infinitely many solutions)

  21. 21. x=32t, y=2+3t, z=t (infinitely many solutions)

  22. 22. x=4t, y=5t, z=t (infinitely many solutions)

  23. 23. y(x)=3 cos 2x+4 sin 2x

  24. 24. y(x)=5 cosh3x+4 sinh3x

  25. 25. y(x)=7e5x+3e5x

  26. 26. y(x)=23e11x+21e11x

  27. 27. y(x)=23e3x+17e5x

  28. 28. y(x)=23e3x8e7x

  29. 29. y(x)=52ex/245ex/3

  30. 30. y(x)=81e4x/340e7x/5

  31. 31. Suggestion: The two lines both pass through the origin.

  32. 32. Suggestion: Two distinct planes in space either are parallel or intersect in a straight line.

  33. 33. (a) No solution (b) A unique solution (c) No solution (d) No solution (e) A unique solution (f) Infinitely many solutions

  34. 34. (a) No solution (b) Infinitely many solutions (c) No solution (d) No solution (e) Infinitely many solutions (f) A unique solution

Section 3.2

  1. 1. x1=1, x2=0, x3=2

  2. 2. x1=5, x2=1, x3=3

  3. 3. x1=13+11t, x2=2+5t, x3=t

  4. 4. x1=35+33t, x2=5+7t, x3=t

  5. 5. x1=13+4t, x2=6+t, x3=5+3t, x4=t

  6. 6. x1=17+t, x2=11+3t, x3=t, x4=4

  7. 7. x1=38s+19t, x2=7+2s7t, x3=s, x4=t

  8. 8. x1=25+10s+22t, x2=s, x3=103t, x4=t

  9. 9. x1=1, x2=3, x3=5, x4=6

  10. 10. x1=63s16t, x2=13s8t, x3=s, x4=5t, x5=t

  11. 11. x1=3, x2=2, x3=4

  12. 12. x1=5, x2=3, x3=2

  13. 13. x1=4+3t, x2=32t, x3=t

  14. 14. x1=5+2t, x2=t, x3=7

  15. 15. Inconsistent—no solution

  16. 16. Inconsistent—no solution

  17. 17. x1=32t, x2=4+t, x3=53t, x4=t

  18. 18. x1=4+2s3t, x2=s, x3=34t, x4=t

  19. 19. x1=3st, x2=5+2s3t, x3=s, x4=t

  20. 20. x1=2+3t, x2=1+s2t, x3=2+2s, x4=s, x5=t

  21. 21. x1=2, x2=1, x3=3, x4=4

  22. 22. x1=3, x2=2, x3=4, x4=1

  23. 23. (a) None (b) k2 (c) k=2

  24. 24. (a) k2 (b) None (c) k=4

  25. 25. (a) k4 (b) k=4 (c) None

  26. 26. (a) All k (b) None (c) None

  27. 27. (a) None (b) k11 (c) k=11

  28. 28. No solution unless c=2a+3b, in which case it has infinitely many solutions.

Section 3.3

  1. 1. [1001]

  2. 2. [1001]

  3. 3. [102013]

  4. 4. [102011]

  5. 5. [105013]

  6. 6. [107016]

  7. 7. [105011000]

  8. 8. [100010001]

  9. 9. [102014000]

  10. 10. [103015000]

  11. 11. [130001000]

  12. 12. [140001000]

  13. 13. [100301020012]

  14. 14. [100401030015]

  15. 15. [101201310000]

  16. 16. [103201430000]

  17. 17. [100230101400125]

  18. 18. [103230145100000]

  19. 19. [102130123100000]

  20. 20. [120230011400000]

  21. 21. x1=3, x2=2, x3=4

  22. 22. x1=5, x2=3, x3=2

  23. 23. x1=4+3t, x2=32t, x3=t

  24. 24. x1=5+2t, x2=t, x3=7

  25. 25. Inconsistent—no solution

  26. 26. Inconsistent—no solution

  27. 27. x1=32t, x2=4+t, x3=53t, x4=t

  28. 28. x1=4+2s3t, x2=s, x3=34t, x4=t

  29. 29. x1=3st, x2=5+2s3t, x3=s, x4=t

  30. 30. x1=2+3t, x2=1+s2t, x3=2+2s, x4=s, x5=t

  31. 31. The sequence 16R3, R25R3, 14R2, R12R2, R13R3 of row operations transforms the first matrix to the identity matrix.

  32. 33. [1001], [1*00],  [0100], [0000]

Section 3.4

  1. 1. [515185]

  2. 2. [16918262215]

  3. 3. [26201262218]

  4. 4. [442220531026352194]

  5. 5. AB=[911012],  BA=[28115]

  6. 6. AB=[7132423104111857],  BA=[1172212167272157]

  7. 7. AB=[26],  BA=[369481251015]

  8. 8. AB=[2115350],BA=[30972013162538]

  9. 9. BA=[4722] but AB is not defined.

  10. 10. AB=[12135631] but BA is not defined.

  11. 11. AB=[11153] but BA is not defined.

  12. 12. Neither product matrix AB nor BA is defined.

  13. 13. A(BC)=(AB)C=[3251217]

  14. 14. A(BC)=(AB)C=[3]

  15. 15. A(BC)=(AB)C=[1215810]

  16. 16. A(BC)=(AB)C=[442291291214181317]

  17. 17. x=s(5, 2, 1, 0)+t(4, 7, 0, 1)

  18. 18. x=s(3, 1, 0, 0)+t(6, 0, 9, 1)

  19. 19. x=s(3, 2, 1, 1, 0)+t(1, 6, 8,  0, 1)

  20. 20. x=s(3, 1, 0, 0, 0)+t(7, 0,  2, 10, 0)

  21. 21. x=r(1, 2, 1, 0, 0)+s(2, 3, 0, 1, 0)+t(7, 4, 2, 0, 1)

  22. 22. x=r(1, 1, 0, 0, 0)+s(7, 0, 1, 1, 0)+t(3, 0, 2, 0, 1)

  23. 23. B=[2132]

  24. 24. B=[7453]

  25. 25. B=[3725]

  26. 29. A2=[a2+bcab+bdac+dcbc+d2], A2+(adbc)I=[a2+adab+bdac+dcd2+ad]=[a(a+d)b(a+d)c(a+d)d(a+d)]=(a+d)A

  27. 31. (a) (A+B)(AB)=[25347134][8454451]=A2B2

  28. 33. [1001], [1001], [+1001], [100+1]

  29. 35. [2121]

  30. 37. [0110]

  31. 39. A(c1x1+c2x2)=c1Ax1+c2Ax2=c1 ·0+c2 ·0=0

  32. 43. A2=[633363336]=3A

Section 3.5

  1. 1. A1=[3243], x=[32]

  2. 2. A1=[5723], x=[2611]

  3. 3. A1=[6756], x=[3328]

  4. 4. A1=[1712175], x=[2510]

  5. 5. A1=12[4253], x=12[87]

  6. 6. A1=13[6734], x=13[2510]

  7. 7. A1=14[7957], x=14[31]

  8. 8. A1=15[101558], x=15[2511]

  9. 9. A1=[5645]

  10. 10. A1=12[6745]

  11. 11. A1=[525212435]

  12. 12. A1=[1827301412]

  13. 13. A1=[13425391271]

  14. 14. A1=[1179433212]

  15. 15. A1=[222727381013]

  16. 16. A1=13[303131132]

  17. 17. A1=14[263221221]

  18. 18. A1=15[122505316]

  19. 19. A1=16[21118952330]

  20. 20. A1=17[310237120]

  21. 21. A1=[0100201010000301]

  22. 22. A1=[1110021201113351]

  23. 23. A1=[4354], X=[7183592345]

  24. 24. A1=[7687], X=[143046163553]

  25. 25. A1=[1194221210], X=[71415132224]

  26. 26. A1=[163116141329], X=[21968321765]

  27. 27. A1=[72017011265], X=[1720241311115674]

  28. 28. A1=[55108815242345], X=[551018815724234513]

Section 3.6

  1. 1. 60

  2. 2. 4

  3. 3. 210

  4. 4. 72

  5. 5. 120

  6. 6. 60

  7. 7. 0

  8. 8. 25

  9. 9. 30

  10. 10. 7

  11. 11. 40

  12. 12. 10

  13. 13. 78

  14. 14. 22

  15. 15. 74

  16. 16. 84

  17. 17. 8

  18. 18. 135

  19. 19. 39

  20. 20. 79

  21. 21. Δ=1, x=10, y=7

  22. 22. Δ=1, x=1, y=1

  23. 23. Δ=1, x=2, y=4

  24. 24. Δ=1, x=5, y=3

  25. 25. Δ=2, x=6, y=3

  26. 26. Δ=2, x=12, y=0

  27. 27. Δ=96, x1=13, x2=23, x3=13

  28. 28. Δ=35, x1=47, x2=37, x3=27

  29. 29. Δ=23, x1=2, x2=3, x3=0

  30. 30. Δ=56, x1=17, x2=914, x3=27

  31. 31. Δ=14, x1=87, x2=107, x3=17

  32. 32. Δ=6, x1=73, x2=9, x3=8

  33. 33. det A=4, A1=14[444161513282523]

  34. 34. det A=35, A1=14[231294191328]

  35. 35. det A=35, A1=135[1525261058152519]

  36. 36. det A=23, A1=123[52017101711148]

  37. 37. det A=29, A1=129[111415171910181514]

  38. 38. det A=6, A1=16[61021521612186]

  39. 39. det A=37, A1=137[21113496659]

  40. 40. det A=107, A1=1107[9121311214152014]

Section 3.7

  1. 1. y(x)=2+3x

  2. 2. y(x)=47x

  3. 3. y(x)=32x2

  4. 4. y(x)=2x+3x2

  5. 5. y(x)=53x+x2

  6. 6. y(x)=107x+2x2

  7. 7. y(x)=13(4x+3x24x3)

  8. 8. y(x)=5+3xx3

  9. 9. y(x)=4+3x+2x2+x3

  10. 10. y(x)=175x+3x22x3

  11. 11. x2+y26x4y12=0, center (3, 2) and radius 5

  12. 12. x2+y2+6x8y75=0, center (3, 4) and radius 10

  13. 13. x2+y2+4x+4y5=0, center (2,2) and radius 13

  14. 14. x2+y210x24y=0, center (5, 12) and radius 13

  15. 15. x2xy+y2=25

  16. 16. 4x27xy+4y2=100

  17. 17. 100x2199xy+100y2=100

  18. 18. 400x2481xy+225y2=3600

  19. 19. y=3+2x

  20. 20. y=10+8x16x2

  21. 21. x2+y2+z22x4y6z155=0, center (1, 2, 3) and radius 13

  22. 22. x2+y2+z210x+14y18z521=0, center (5, 7, 9) and radius 26

  23. 23. P(t)=49.0610.0722t+0.00798t2

  24. 24. P(t)=56.590+0.30145t0.007375t2

  25. 25. P(t)=62.813+1.37915t0.012375t2

  26. 26. P(t)=34.838+0.7692t+0.0064t2

  27. 27. P(t)=44.678+0.850417t0.05105t2+0.000983833t3

  28. 28. P(t)=51.619+0.672433t0.019565t2+0.000203167t3

  29. 29. P(t)=54.973+0.308667t+0.059515t20.00119817t3

  30. 30. P(t)=28.053+0.592233t+0.00907t20.0000443333t3

  31. 31. P(t)=39.478+0.209692t+0.0564163t20.00292992t3+0.0000391375t4

  32. 32. P(t)=44.461+0.7651t0.000489167t20.000516t3+7.19167×106t4

  33. 33. P(t)=47.197+1.22537t0.077192t2+0.00373475t30.0000493292t4

  34. 34. P(t)=20.190+1.00003t0.031775t2+0.00116067t30.000041205t4

  35. 36. y=2x26x+7

  36. 38. x2+y2+6x8y75=0, center (3, 4) and radius 10

  37. 40. 400x2481xy+225y23600=0

Chapter 4

Section 4.1

  1. 1. 51,  (5, 8, 11),  (2, 23, 0)

  2. 2. 9, (1, 4, 1), (15, 16, 26)

  3. 3. 321, 9i3j+3k, 14i21j+43k

  4. 4. 17, 4ij3k, 6i7j+12k

  5. 5. Linearly dependent

  6. 6. Linearly dependent

  7. 7. Linearly dependent

  8. 8. Linearly dependent

  9. 9. w=3u+2v

  10. 10. w=2u3v

  11. 11. w=u2v

  12. 12. w=3u+5v

  13. 13. w=2u3v

  14. 14. w=7u+5v

  15. 15. Linearly dependent

  16. 16. Linearly dependent

  17. 17. Linearly dependent

  18. 18. Linearly dependent

  19. 19. Linearly dependent; a=3, b=2, c=1

  20. 20. Linearly dependent; a=2, b=3, c=1

  21. 21. Linearly dependent; a=11, b=4, c=1

  22. 22. Linearly dependent

  23. 23. Linearly dependent

  24. 24. Linearly dependent

  25. 25. t=2uv+3w

  26. 26. t=u+5vw

  27. 27. t =2u+6v+w

  28. 28. t =u+v+w

Section 4.2

  1. 1. W is a subspace of R3.

  2. 2. W is a subspace of R3.

  3. 3. W is not a subspace of R3.

  4. 4. W is not a subspace of R3.

  5. 5. W is a subspace of R4.

  6. 6. W is a subspace of R4.

  7. 7. W is not a subspace of R2.

  8. 8. W is a subspace of R2.

  9. 9. W is not a subspace of R2.

  10. 10. W is not a subspace of R2.

  11. 11. W is a subspace of R4.

  12. 12. W is not a subspace of R4.

  13. 13. W is not a subspace of R4.

  14. 14. W is not a subspace of R4.

  15. 15. u=(1, 0,1,0) and v=(4, 2, 0, 1)

  16. 16. u=(1, 1, 0, 1) and v=(5, 3, 0, 1)

  17. 17. u=(1, 3, 2, 0) and v=(2, 1, 0, 1)

  18. 18. u=(2, 1, 2, 1, 0) and v=(3, 4, 5, 0, 1)

  19. 19. u=(1, 2, 1, 0)

  20. 20. u=(5, 3, 2, 1)

  21. 21. u=(3, 2,4, 1)

  22. 22. u=(6, 4,3, 1)

Section 4.3

  1. 1. Linearly dependent

  2. 2. Linearly dependent

  3. 3. Linearly dependent

  4. 4. Linearly dependent

  5. 5. Linearly dependent

  6. 6. Linearly dependent

  7. 7. Linearly dependent

  8. 8. Linearly dependent

  9. 9. w=2v13v2

  10. 10. w=7v1+4v2

  11. 11. w=v12v2

  12. 12. w=2v1+5v2

  13. 13. w cannot be expressed as a linear combination of v1 and v2

  14. 14. w cannot be expressed as a linear combination of v1, v2 and v3

  15. 15. w=3v12v2+4v3

  16. 16. w=6v12v2+3v3

  17. 17. The vectors v1, v2, v3 are linearly independent.

  18. 18. 3v1+v2+5v3=0

  19. 19. The vectors v1, v2, v3 are linearly independent.

  20. 20. The vectors v1, v2, v3 are linearly independent.

  21. 21. v12v2v3=0

  22. 22. 7v1+5v29v3=0

Section 4.4

  1. 1. The vectors v1 and v2 form a basis for R2.

  2. 2. The vectors v1, v2, v3 do not form a basis for R3.

  3. 3. The given vectors do not form a basis for R3.

  4. 4. The given vectors do not form a basis for R4.

  5. 5. The three vectors v1, v2, v3 do not form a basis for R3.

  6. 6. The four given vectors form a basis for R3.

  7. 7. The three given vectors form a basis for R3.

  8. 8. The three given vectors form a basis for R4.

  9. 9. The plane x2y+5z=0 is a 2-dimensional subspace of R3 with basis consisting of the vectors v1=(2, 1, 0) and v2=(5, 0, 1).

  10. 10. The plane yz=0 is a 2-dimensional subspace of R3 with basis consisting of the vectors v1=(1, 0, 0) and v2=(0, 1, 1).

  11. 11. The line is a 1-dimensional subspace of R3 with basis consisting of the vector v=(3, 1, 1).

  12. 12. Hence the subspace consisting of all such vectors is 3-dimensional with basis consisting of the vectors v1=(1, 1, 0, 0), v2=(1, 0, 1, 0), and v3=(1, 0, 0, 1).

  13. 13. The subspace consisting of all such vectors is 2-dimensional with basis consisting of the vectors v1=(3, 0, 1, 0) and v2=(0, 4, 0, 1).

  14. 14. The subspace consisting of all such vectors is 2-dimensional with basis consisting of the vectors v1=(2, 1, 0, 0) and v2=(0, 0,3, 1).

  15. 15. The solution space of the given system is 1-dimensional with basis consisting of the vector v1=(11, 7, 1).

  16. 16. The solution space of the given system is 1-dimensional with basis consisting of the vector v1=(11,5, 1).

  17. 17. The solution space of the given system is 2-dimensional with basis consisting of the vectors v1=(11,3, 1, 0) and v2(11,5, 0, 1).

  18. 18. The solution space of the given system is 2-dimensional with basis consisting of the vectors v1=(3, 1, 0, 0) and v2(25, 0, 5, 1).

  19. 19. The solution space of the given system is 2-dimensional with basis consisting of the vectors v1=(3,2, 1, 0) and v2=(4,3, 0, 1).

  20. 20. The solution space of the given system is 2-dimensional with basis consisting of the vectors v1=(1,3, 1, 0) and v2=(2, 1, 0, 1).

  21. 21. The solution space of the given system is 2-dimensional with basis consisting of the vectors v1=(1,1, 1, 0) and v2=(5,3, 0, 1).

  22. 22. The solution space of the given system is 2-dimensional with basis consisting of the vectors v1=(2, 1, 0, 0) and v2=(5, 0,7, 1).

  23. 23. The solution space of the given system is 1-dimensional with basis consisting of the vector v1=(2,3, 1, 0).

  24. 24. The solution space of the given system is 3-dimensional with basis consisting of the vectors v1=(2, 2, 1, 0, 0), v2=(1, 3, 0, 1, 0), and v3=(3,1, 0, 0, 1).

  25. 25. The solution space of the given system is 3-dimensional with basis consisting of the vectors v1=(2, 1, 0, 0, 0), v2=(2, 0, 1, 1, 0), and v3=(3, 0,4, 0, 1).

  26. 26. The solution space of the given system is 2-dimensional with basis consisting of the vectors v1=(2, 1, 2, 1, 0) and v2=(3,4, 5, 0, 1).

Section 4.5

  1. 1. Row basis: The first and second row vectors of E. Column basis: The first and second column vectors of A.

  2. 2. Row basis: The first and second row vectors of E. Column basis: The first and second column vectors of A.

  3. 3. Row basis: The first and second row vectors of E. Column basis: The first and second column vectors of A.

  4. 4. Row basis: The three row vectors of E. Column basis: The first three column vectors of A.

  5. 5. Row basis: The three row vectors of E. Column basis: The first, second, and fourth column vectors of A.

  6. 6. Row basis: The three row vectors of E. Column basis: The first, second, and fourth column vectors of A.

  7. 7. Row basis: The first two row vectors of E. Column basis: The first two column vectors of A.

  8. 8. Row basis: The first three row vectors of E. Column basis: The first, second, and fourth column vectors of A.

  9. 9. Row basis: The first three row vectors of E. Column basis: The first three column vectors of A.

  10. 10. Row basis: The first three row vectors of E. Column basis: The first, second, and fourth column vectors of A.

  11. 11. Row basis: The first three row vectors of E. Column basis: The first, second, and fifth column vectors of A.

  12. 12. Row basis: The first three row vectors of E. Column basis: The first, second, and fifth column vectors of A.

  13. 13. Linearly independent: v1 and v2

  14. 14. Linearly independent: v1 and v2

  15. 15. Linearly independent: v1, v2, and v4

  16. 16. Linearly independent: v1, v2, v4, and v5

  17. 17. Basis vectors: v1, v2, e2

  18. 18. Basis vectors: v1, v2, e2

  19. 19. Basis vectors: v1, v2, e2, e3

  20. 20. Basis vectors: v1, v2, e1, e3

  21. 21. The first and second equations are irredundant.

  22. 22. The first and second equations are irredundant.

  23. 23. The first, second, and fourth equations are irredundant.

  24. 24. The first, second, and fifth equations are irredundant.

Section 4.6

  1. 1. Yes, the three vectors are mutually orthogonal.

  2. 2. Yes, the three vectors are mutually orthogonal.

  3. 3. Yes, the three vectors are mutually orthogonal.

  4. 4. Yes, the three vectors are mutually orthogonal.

  5. 5. a2=7, b2=7, c2=14

  6. 6. a2=18, b2=18, c2=36

  7. 7. a2=19, b2=25, c2=44

  8. 8. a2=103, b2=112, c2=215

  9. 9. A=B=45°

  10. 10. A=B=45°

  11. 11. A=41.08°, B=48.92°

  12. 12. A=43.80°, B=46.20°

  13. 13. u1=(2, 1, 0), u2=(3, 0, 1)

  14. 14. u1=(5, 1, 0), u2=(3, 0, 1)

  15. 15. u1=(2, 1, 0, 0), u2=(3, 0, 1, 0), u3=(5, 0, 0, 1)

  16. 16. u1=(7, 1, 0, 0), u2=(6, 0, 1, 0), u3=(9, 0, 0, 1)

  17. 17. u1=(7,3, 1, 0), u2=(19, 5, 0, 1)

  18. 18. u1=(12,3, 1, 0), u2=(16, 7, 0, 1)

  19. 19. u1=(13, 4, 1, 0, 0), u2=(4,3, 0, 1, 0), u3=(11, 4, 0, 0, 1)

  20. 20. u1=(5, 1, 1, 0, 0), u2=(12, 4, 0, 1, 0), u3=(19, 7, 0, 0, 1)

  21. 21. u1=(1,1, 1, 0, 0), u2=(0,1, 0,1, 1)

  22. 22. u1=(2, 1, 1, 0, 0), u2=(1,2, 0, 1, 0)

Section 4.7

  1. 1. It is a subspace.

  2. 2. It is a subspace.

  3. 3. It is a not subspace.

  4. 4. It is a not subspace.

  5. 5. It is a subspace.

  6. 6. It is a not subspace.

  7. 7. It is a not subspace.

  8. 8. It is a subspace.

  9. 9. It is a not subspace.

  10. 10. It is a subspace.

  11. 11. It is a subspace.

  12. 12. It is a not subspace.

  13. 13. The functions sin x and cos x are linearly independent.

  14. 14. The funtions ex and xex are linearly independent.

  15. 15. The functions (1+x), (1x), and (1x2) are linearly independent.

  16. 16. The three given polynomials are linearly dependent.

  17. 17. The three given trigonometric functions are linearly dependent.

  18. 18. The two given linear combinations of sin x and cos x are linearly independent.

  19. 19. A=3 and B=2.

  20. 20. A=2 and B=C=1.

  21. 21. A=2, B=2, and C=0.

  22. 22. A=1, B=4, and C=3.

  23. 23. The solution space is 3-dimensional with basis {1, x, x2}.

  24. 24. The solution space is 4-dimensional with basis {1, x, x2, x3}.

  25. 25. The solution space is 2-dimensional with basis {1, e5x}.

  26. 26. The solution space is 2-dimensional with basis {1, e10x}.

Chapter 5

Section 5.1

  1. 1. y(x)=52ex52ex

  2. 2. y(x)=2e3x3e3x

  3. 3. y(x)=3 cos 2x+4 sin 2x

  4. 4. y(x)=10 cos 5x2 sin 5x

  5. 5. y(x)=2exe2x

  6. 6. y(x)=4e2x+3e3x

  7. 7. y(x)=68ex

  8. 8. y(x)=13(142e3x)

  9. 9. y(x)=2ex+xex

  10. 10. y(x)=3e5x2xe5x

  11. 11. y(x)=5exsin x

  12. 12. y(x)=e3x(2 cos 2x+3 sin 2x)

  13. 13. y(x)=5x2x2

  14. 14. y(x)=3x216/x3

  15. 15. y(x)=7x5xln x

  16. 16. y(x)=2 cos(ln x)+3 sin(ln x)

  17. 21. Linearly independent

  18. 22. Linearly independent

  19. 23. Linearly independent

  20. 25. Linearly dependent

  21. 25. Linearly independent

  22. 26. Linearly independent

  23. 28. y(x)=12 cos xsin x

  24. 29. There is no contradiction because if the given differential equation is divided by x2 to get the form in Eq. (8), then theresulting coefficient functions p(x)=4/x and q(x)=6/x2 are not continuous at x=0.

  25. 33. y(x)=c1ex+c2e2x

  26. 34. y(x)=c1e5x+c2e3x

  27. 35. y(x)=c1+c2e5x

  28. 36. y(x)=c1+c2e3x/2

  29. 37. y(x)=c1ex/2+c2ex

  30. 38. y(x)=c1ex/2+c2e3x/2

  31. 39. y(x)=(c1+c2x)ex/2

  32. 40. y(x)=(c1+c2x)e2x/3

  33. 41. y(x)=c1e4x/3+c2e5x/2

  34. 42. y(x)=c1e4x/7+c2e3x/5

  35. 43. y+10y=0

  36. 44. y100y=0

  37. 45. y+20y+100y=0

  38. 46. y110y+1000y=0

  39. 47. y=0

  40. 48. y2yy=0

  41. 49. The high point is (ln 74, 167).

  42. 50. (ln 2, 2)

  43. 52. y(x)=c1x+c2/x

  44. 53. y(x)=c1x4+c2x3

  45. 54. y(x)=c1x3/2+c2x1/2

  46. 55. y(x)=c1+c2 ln x

  47. 56. y(x)=x2(c1+c2 ln x)

Section 5.2

  1. 1. 15·(2x)16·(3x2)6·(5x8x2)0

  2. 2. (4)(5)+(5)(23x2)+(1)(10+15x2)0

  3. 3. 1·0+0· sin x+0·ex0

  4. 4. (6)(17)+(51)(2 sin2 x)+(34)(3 cos2 x)0

  5. 5. 1·1734·cos2 x+17·cos 2x0

  6. 6. (1)(ex)+(1)(coshx)+(1)(sinhx)0

  7. 13. y(x)=43ex13e2x

  8. 14. y(x)=12(3ex6e2x+3e3x)

  9. 15. y(x)=(22x+x2)ex

  10. 16. y(x)=12ex+13e2x10xe2x

  11. 17. y(x)=19(292 cos 3x3 sin 3x)

  12. 18. y(x)=ex(2 cos x sin x)

  13. 19. y(x)=x+2x2+3x3

  14. 20. y(x)=2xx2+x2 ln x

  15. 21. y(x)=2 cos x5 sin x+3x

  16. 22. y(x)=4e2xe2x3

  17. 23. y(x)=ex+4e3x2

  18. 24. y(x)=ex(3 cos x+4 sin x)+x+1

  19. 38. y2(x)=1x3

  20. 39. y2(x)=xex/2

  21. 40. y2(x)=xex

  22. 41. y2(x)=x+2

  23. 42. y2(x)=1+x2

Section 5.3

  1. 1. y(x)=c1e2x+c2e2x

  2. 2. y(x)=c1+c2e3x/2

  3. 3. y(x)=c1e2x+c2e5x

  4. 4. y(x)=c1ex/2+c2e3x

  5. 5. y(x)=c1e3x+c2xe3x

  6. 6. y(x)=e5x/2[c1exp(12x5)+c2exp(12x5)]

  7. 7. y(x)=c1e3x/2+c2xe3x/2

  8. 8. y(x)=e3x(c1 cos 2x+c2 sin 2x)

  9. 9. y(x)=e4x(c1 cos 3x+c2 sin 3x)

  10. 10. y(x)=c1+c2x+c3x2+c4e3x/5

  11. 11. y(x)=c1+c2x+c3e4x+c4xe4x

  12. 12. y(x)=c1+c2ex+c3xex+c4x2ex

  13. 13. y(x)=c1+c2e2x/3+c3xe2x/3

  14. 14. y(x)=c1ex+c2ex+c3 cos 2x+c4 sin 2x

  15. 15. y(x)=c1e2x+c2xe2x+c3e2x+c4xe2x

  16. 16. y(x)=(c1+c2x) cos 3x+(c3+c4x) sin 3x

  17. 17. y(x)=c1 cos (x/2)+c2 sin (x/2)+c3 cos (2x/3)+c4 sin (2x/3)

  18. 18. y(x)=c1e2x+c2e2x+c3 cos 2x+c4 sin 2x

  19. 19. y(x)=c1ex+c2ex+c3xex

  20. 20. y(x)=ex/2[(c1+c2x) cos (12x3)+(c3+c3x) sin (12x3)]

  21. 21. y(x)=5ex+2e3x

  22. 22. y(x)=ex/3[3 cos (x/3)+53 sin (x/3)]

  23. 23. y(x)=e3x(3 cos 4x2 sin 4x)

  24. 24. y(x)=12(7+e2x+8ex/2)

  25. 25. y(x)=14(13+6x+9e2x/3)

  26. 26. y(x)=15(249e5x25xe5x)

  27. 27. y(x)=c1ex+c2e2x+c3xe2x

  28. 28. y(x)=c1e2x+c2ex+c3ex/2

  29. 29. y(x)=c1e3x+e3x/2[c2 cos (32x3)+c3 sin (32x3)]

  30. 30. y(x)=c1ex+c2e2x+c3 cos (x3)+c4 sin (x3)

  31. 31. y(x)=c1ex+e2x(c2 cos 2x+c3 sin 2x)

  32. 32. y(x)=c1e2x+(c2+c3x+c4x2)ex

  33. 33. y(x)=c1e3x+e3x(c2 cos 3x+c3 sin 3x)

  34. 34. y(x)=c1e2x/3+c2 cos 2x+c3 sin 2x

  35. 35. y(x)=c1ex/2+c2ex/3+c3 cos 2x+c4 sin 2x

  36. 36. y(x)=c1e7x/9+ex(c2 cos x+c3 sin x)

  37. 37. y(x)=11+5x+3x2+7ex

  38. 38. y(x)=2e5x2 cos 10x

  39. 39. y(3)6y+12y8y=0

  40. 40. y(3)2y+4y8y=0

  41. 41. y(4)16y=0

  42. 42. y(6)+12y(4)+48y+64y=0

  43. 44. (a) x=i, 2i (b) x=i, 3i

  44. 45. y(x)=c1eix+c2e3ix

  45. 46. y(x)=c1e3ix+c2e2ix

  46. 47. y(x)=c1exp([1+i3]x)+c2exp([1+i3]x)

  47. 48. y(x)=13(ex+exp[12(1+i3)x]+exp[12(1i3)x])

  48. 49. y(x)=2e2x5ex+3 cos x9 sin x

  49. 52. y(x)=c1 cos (3 ln x)+c2 sin (3 ln x)

  50. 53. y(x)=x3[c1 cos (4 ln x)+c2 sin (4 ln x)]

  51. 54. y(x)=c1+c2 ln x+c3x3

  52. 55. y(x)=c1+x2(c2+c3 ln x)

  53. 56. y(x)=c1+c2 ln x+c3(ln x)2

  54. 57. y(x)=c1+x3(c2x3+c3x+3)

  55. 58. y(x)=x1[c1+c2 ln x+c3(ln x)2]

Section 5.4

  1. 1. Frequency: 2 rad/s (1/π HZ); period: π s

  2. 2. Frequency: 8 rad/sec (4/π HZ); period: π/4 sec

  3. 3. Amplitude: 2m; frequency: 5 rad/s; period: 2π/5 s

  4. 4. (a) x(t)=1312cos (12tα) with α=2πtan1(5/12)5.8884; (b) Amplitude: 1312 m; period: π/6 sec

  5. 6. About 7.33 mi

  6. 7. About 10450 ft

  7. 8. 29.59 in.

  8. 10. Amplitude: 100 cm; period: about 2.01 sec

  9. 11. About 3.8 in.

  10. 13. (a) x(t)=50(e2t/5et/2); (b) 4.096 exactly

  11. 14. (a) x(t)=25et/5 cos (3tα) with α=tan1(3/4)0.6435; (b) envelope curves x=±25et/5; pseudoperiod 2π/3

  12. 15. x(t)=4e2t2e4t, overdamped; u(t)=2 cos (22t)

  13. 16. x(t)=4e3t2e7t, overdamped; u(t)22221 cos (21t0.2149)

  14. 17. x(t)=5e4t(2t+1), critically damped; u(t)525 cos (4t5.8195)

  15. 18. x(t)=2e3t cos (4t3π2), underdamped; u(t)=85cos (5t3π2)

  16. 19. x(t)13313e5t/2 cos (6t0.8254), underdamped; u(t)413233 cos (132t0.5517)

  17. 20. x(t)13e4t cos (2t1.1760), underdamped; u(t)1295 cos (25t0.1770)

  18. 21. x(t)10e5t cos (10t0.9273), underdamped; u(t)214 cos (55t0.6405)

  19. 22. (b) The time-varying amplitude is 233, the frequencyis 43 rad/s, and the phase angle is π/6.

  20. 23. (a) k7018 lb/ft; (b) After about 2.47 s

  21. 34. Damping constant: c11.51 lb/ft/s; spring constant: k189.68 lb/ft

Section 5.5

  1. 1. yp(x)=125e3x

  2. 2. yp(x)=14(5+6x)

  3. 3. yp(x)=139(cos 3x5 sin 3x)

  4. 4. yp(x)=19(4ex+3xex)

  5. 5. yp(x)=126(13+3 cos 2x2 sin 2x)

  6. 6. yp(x)=1343(456x+49x2)

  7. 7. yp(x)=16(exex)=13sinhx

  8. 8. yp(x)=14xsinh2x

  9. 9. yp(x)=13+116(2x2x)ex

  10. 10. yp(x)=16(2x sin 3x3x cos 3x)

  11. 11. yp(x)=18(3x22x)

  12. 12. yp(x)=2x+12xsin x

  13. 13. yp(x)=165ex(7 sin x4 cos x)

  14. 14. yp(x)=124(3x2ex+x3ex)

  15. 15. yp(x)17

  16. 16. yp(x)=181(45+e3x6xe3x+9x2e3x)

  17. 17. yp(x)=14(x2 sin xx cos x)

  18. 18. yp(x)=1144(24xex19xe2x+6x2e2x)

  19. 19. yp(x)=18(10x24x3+x4)

  20. 20. yp(x)=7+13xex

  21. 21. yp(x)=xex(A cos x+B sin x)

  22. 22. yp(x)=Ax3+Bx4+Cx5+Dxex

  23. 23. yp(x)=Ax cos 2x+Bx sin 2x+Cx2 cos 2x+Dx2 sin 2x

  24. 24. yp(x)=Ax+Bx2+(Cx+Dx2)e3x

  25. 25. yp(x)=Axex+Bx2ex+Cxe2x+Dx2e2x

  26. 26. yp(x)=(Ax+Bx2)e3x cos 2x+(Cx+Dx2)e3x sin 2x

  27. 27. yp(x)=Ax cos x+Bx sin x+Cx cos 2x+Dx sin 2x

  28. 28. yp(x)=(Ax+Bx2+Cx3) cos 3x+(Dx+Ex2+Fx3) sin 3x

  29. 29. yp(x)=Ax3ex+Bx4ex+Cxe2x+Dxe2x

  30. 30. yp(x)=(A+Bx+Cx2) cos x+(D+Ex+Fx2) sin x

  31. 31. y(x)=cos 2x+34sin 2x+12x

  32. 32. y(x)=16(15ex16e2x+ex)

  33. 33. y(x)=cos 3x215sin 3x+15sin 2x

  34. 34. y(x)=cos x sin x+12x sin x

  35. 35. y(x)=ex(2 cos x52sin x)+12x+1

  36. 36. y(x)=1192(234+240x9e2x33e2x12x24x4)

  37. 37. y(x)=44ex+3xex+x12x2ex+16x3ex

  38. 38. y(x)=185[ex(176 cos x+197 sin x)(6 cos 3x+7 sin 3x)]

  39. 39. y(x)=3+3x12x2+16x3+4ex+xex

  40. 40. y(x)=14(5ex+5ex+10 cos x20)

  41. 41. yp(x)=255450x+30x2+20x3+10x44x5

  42. 42. y(x)=10ex+35e2x+210 cos x+390 sin x+yp(x) where yp(x) is the particular solution of Problem 41 .

  43. 43. (b) y(x)=c1 cos 2x+c2 sin 2x+14cos x120cos 3x

  44. 44. y(x)=ex/2[c1 cos (12x3)+c2 sin (12x3)]+126(3 cos 2x+2 sin 2x)+1482(15 cos 4x+4 sin 4x)

  45. 45. y(x)=c1 cos 3x+c2 sin 3x+124110cos 2x156cos 4x

  46. 46. y(x)=c1 cos x+c2 sin x+116(3x cos x+3x2 sin x)+1128(3 sin 3x4x cos 3x)

  47. 47. yp(x)=23ex

  48. 48. yp(x)=112(6x+1)e2x

  49. 49. yp(x)=x2e2x

  50. 50. yp(x)=116(4x cosh 2xsinh 2x)

  51. 51. yp(x)=14(cos 2x cos x sin 2x sin x)+120(cos 5x cos 2x+ sin 5x sin 2x)=15cos 3x (!)

  52. 52. yp(x)=16x cos 3x

  53. 53. yp(x)=23x sin 3x+29(cos 3x) ln |cos 3x|

  54. 54. yp(x)=1(cos x) ln | csc x cot x|

  55. 55. yp(x)=18(1xsin 2x)

  56. 56. yp(x)=19ex(3x+2)

  57. 58. yp(x)=x3(ln x1)

  58. 59. yp(x)=14x4

  59. 60. yp(x)=725x4/3

  60. 61. yp(x)=ln x

  61. 62. yp(x)=x2+x ln |1+x1x|+12(1+x2) ln |1x2|

Section 5.6

  1. 1. x(t)=2 cos 2t2 cos 3t

  2. 2. x(t)=32sin 2t sin 3t

  3. 3. x(t)=138388 cos (10tα)+5 cos (5tβ) with α=2πtan1(1/186)6.2778 and β=tan1(4/3)0.9273.

  4. 4. x(t)=2106 cos (5tα)+10 cos 4t with α=πtan1(9/5)2.0779

  5. 5. x(t)=(x0C) cos ω0t+C cos ωt, where C=F0/(kmω2)

  6. 7. xsp(t)=1013cos (3tα) with α=πtan1(12/5)1.9656

  7. 8. xsp(t)=425cos (5tα) with α=2πtan1(3/4)5.6397

  8. 9. xsp(t)=340001cos (10tα) with α=π+tan1(199/20)4.6122

  9. 10. xsp(t)=1079361 cos (10tα) with α=π+tan1(171/478)3.4851

  10. 11. xsp(t)=104cos (3tα) with α=πtan1(3)1.8925 xtr(t)=542e2t cos (tβ) with β=2πtan1(7)4.8543

  11. 12. xsp(t)=5329cos (3tα) with α=π+tan1(2/5)3.5221 xtr(t)=25629e3t cos (2tβ) with β=tan1(5/2)1.1903

  12. 13. xsp(t)=3001469cos (10tα) with α=πtan1(10/37)2.8776 xtr(t)=21133141469et cos (5tβ) with β=2πtan1(421/12895)6.2505

  13. 14. xsp(t)=485 cos (tα) with α=tan1(22)1.5254xtr(t)=3665e4t cos (3tβ) with β=π+tan1(52/31)4.1748

  14. 15. C(ω)=2/4+ω4; there is no practical resonance frequency.

  15. 16. C(ω)=10/25+6ω2+ω4; there is no practical resonance frequency.

  16. 17. C(ω)=50/202554ω2+ω4; there is practical resonance at frequency ω=33.

  17. 18. C(ω)=100/4225001200ω2+ω4; there is practical resonance at frequency ω=106.

  18. 19. ω=384 rad/sec (approximately 3.12 Hz)

  19. 20. ω44.27 rad/sec (approximately 422.75 rpm)

  20. 21. ω0=(g/L)+(k/m)

  21. 22. ω0=k/(m+I/a2)

  22. 23. (a) Natural frequency: 10 rad/s (approximately 0.50 Hz); (b) amplitude: approximately 10.625 in.

Chapter 6

Section 6.1

  1. 1. λ1=2: v1=(1, 1); λ2=3: v2=(2, 1)

  2. 2. λ1=1: v1=(1, 1); λ2=2: v2=(2, 1)

  3. 3. λ1=2: v2=(1, 1); λ2=5: v1=(2, 1)

  4. 4. λ1=1: v2=(1, 1); λ2=2: v1=(3, 2)

  5. 5. λ1=1: v2=(1, 1); λ2=4: v1=(3, 2)

  6. 6. λ1=2: v2=(1, 1); λ2=3: v1=(4, 3)

  7. 7. λ1=2: v2=(1, 1); λ2=4: v1=(4, 3)

  8. 8. λ1=2: v2=(2, 3); λ2=1: v1=(3, 4)

  9. 9. λ1=3: v1=(2, 1); λ2=4: v2=(5, 2)

  10. 10. λ1=4: v1=(2, 1); λ2=5: v2=(5, 2)

  11. 11. λ1=4: v2=(2, 3); λ2=5: v1=(5, 7)

  12. 12. λ1=3: v2=(3, 2); λ2=4: v1=(5, 3)

  13. 13. λ1=0: v1=(0,1, 2); λ2=1: v2=(0,1, 3); λ3=2: v3=(1, 0, 2)

  14. 14. λ1=0: v1=(0,1, 2); λ2=2: v2=(0,1, 3); λ3=5: v3=(1, 0, 2)

  15. 15. λ1=0: v1=(1, 1, 0); λ2=1: v2=(2, 1, 1); λ3=2: v3=(1, 0, 2)

  16. 16. λ1=0: v1=(1, 1, 1); λ2=1: v2=(1, 1, 0); λ3=3: v3=(1, 0, 2)

  17. 17. λ1=1: v1=(1, 0, 1); λ2=2: v2=(1, 1, 2); λ3=3: v3=(1, 0, 0)

  18. 18. λ1=1: v1=(1, 0, 3); λ2=2: v2=(0,1, 3); λ3=3: v3=(0,2, 5)

  19. 19. λ1=λ2=1: v1=(1, 0, 1), v2=(3, 1, 0); λ3=3, v3=(1, 0, 0)

  20. 20. λ1=λ2=1: v1=(1, 0, 2), v2=(3, 2, 0); λ3=2, v3=(0,2, 5)

  21. 21. λ1=1: v1=(1, 1, 0); λ2=λ3=2: v2=(3, 2, 0), v3=(1, 0, 2)

  22. 22. λ1=λ2=1: v1=(1, 1, 0), v2=(1, 0, 2); λ3=2, v3=(1, 1, 1)

  23. 23. λ1=1: v1=(1, 0, 0, 0); λ2=2: v2=(2, 1, 0, 0); λ3=3: v3=(3, 2, 1, 0); λ4=4: v4=(4, 3, 2, 1)

  24. 24. λ1=λ2=1: v1=(1, 0, 0, 0), v2=(0, 1, 0, 0); λ3=λ4=3: v3=(0, 0, 0, 1), v4=(2, 2, 1, 0)

  25. 25. λ1=λ2=1: v1=(1, 0, 0, 0), v2=(0, 1, 0, 0); λ3=λ4=2: v3=(0, 0, 0, 1), v4=(1, 1, 1, 0)

  26. 26. λ1=2: v1=(1, 0, 0, 2); λ2=1: v2=(0, 0, 1, 0); λ3=1: v3=(1, 0, 0, 1); λ4=2: v4=(0, 1, 0, 0)

  27. 27. λ1=i: v1=(+i, 1); λ2=+i: v2=(i, 1)

  28. 28. λ1=6i: v1=(i, 1); λ2=+6i: v2=(+i, 1)

  29. 29. λ1=6i: v1=(i, 2); λ2=+6i: v2=(+i, 2)

  30. 30. λ1=12i: v1=(i, 1); λ2=+12i: v2=(+i, 1)

  31. 31. λ1=12i: v1=(+2i, 1); λ2=+12i: v2=(2i, 1)

  32. 32. λ1=12i: v1=(i, 3); λ2=+12i: v2=(+i, 3)

  33. 40. We find that Tr A=12 and detA=60, so the characteristic polynomial of the given matrix A is p(λ)=λ3+12λ2+c1λ+60. Eigenvalues and eigenvectors: λ1=3: v1=(3, 2, 1); λ2=4: v2=(5, 7, 7); λ3=5: v3=(1, 1, 2)

  34. 41. We find that Tr A=8 and detA=60, so the characteristic polynomial of the given matrix A is p(λ)=λ48λ3+c2λ2+c1λ60. Eigenvalues and eigenvectors: λ1=2: v1=(1, 0, 1, 2); λ2=2: v2=(3, 4, 0, 3); λ3=3: v3=(3, 1, 2, 4); λ4=5: v4=(1, 1, 0, 1)

Section 6.2

  1. 1. λ1=1, λ2=3; P=[1211], D=[1003]

  2. 2. λ1=0, λ2=2; P=[1312], D=[0002]

  3. 3. λ1=2, λ2=3; P=[1312], D=[2003]

  4. 4. λ1=1, λ2=2; P=[1413], D=[1002]

  5. 5. λ1=1, λ2=3; P=[1413], D=[1003]

  6. 6. λ1=1, λ2=2; P=[2334], D=[1002]

  7. 7. λ1=1, λ2=2; P=[2512], D=[1002]

  8. 8. λ1=1, λ2=2; P=[3523], D=[1002]

  9. 9. The double eigenvalue λ1=λ2=1 has only the single associated eigenvector v1=(2, 1), so the matrix A is not diagonalizable.

  10. 10. The double eigenvalue λ1=λ2=2 has only the single associated eigenvector v1=(1, 1), so the matrix A is not diagonalizable.

  11. 11. The double eigenvalue λ1=λ2=2 has only the single associated eigenvector v1=(1, 3), so the matrix A is not diagonalizable.

  12. 12. The double eigenvalue λ1=λ2=1 has only the single associated eigenvector v1=(3, 4), so the matrix A is not diagonalizable.

  13. 13. λ1=1, λ2=λ3=2; P=[103001010], D=[100020002]

  14. 14. λ1=λ2=0, λ3=1;P=[111011210], D=[000000001]

  15. 15. λ1=0, λ2=λ3=1; P=[113102020], D=[000010001]

  16. 16. λ1=λ2=1,  λ3=3; P=[011010102], D=[100010003]

  17. 17. λ1=1, λ2=1, λ3=2; P=[111101021], D=[100010002]

  18. 18. λ1=1, λ2=2,  λ3=3; P=[111101021], D=[100020003]

  19. 19. λ1=1, λ2=2, λ3=3; P=[111110102], D=[100020003]

  20. 20. λ1=2, λ2=5, λ3=6; P=[100012335], D=[200050006]

  21. 21. The triple eigenvalue λ1=λ2=λ3=1 has only the two associated eigenvectors v1=(0, 0, 1) and v2=(1, 1, 0), so the matrix A is not diagonalizable.

  22. 22. The triple eigenvalue λ1=λ2=λ3=1 has only the single associated eigenvector v1=(1, 1, 1), so the matrix A is not diagonalizable.

  23. 23. The eigenvalues λ1=λ2=1 and λ3=2 have only the two associated eigenvectors v1=(1, 1, 1) and v3=(1, 1, 0), so the matrix A is not diagonalizable.

  24. 24. The eigenvalues λ1=1 and λ2=λ3=2 have only the two associated eigenvectors v1=(1, 1, 0) and v2=(1, 1, 1), so the matrix A is not diagonalizable.

  25. 25. λ1=λ2=1, λ3=λ4=1; P=[0101011001001000], D=[1000010000100001]

  26. 26. λ1=λ2=λ3=1, λ4=2; P=[0011010000010001], D=[1000010000100002]

  27. 27. The eigenvalues λ1=λ2=λ3=1 and λ4=2 have only the two associated eigenvectors v1=(1, 0, 0, 0) and v4=(1, 1, 1, 1), so the matrix A is not diagonalizable.

  28. 28. The eigenvalues λ1=λ2=1 and λ3=λ4=2 have only the two associated eigenvectors v1=(1, 0, 0, 0) and v3=(1, 1, 1, 0), so the matrix A is not diagonalizable.

Section 6.3

  1. 1. P=[1211], D=[1002], A5=[63623130]

  2. 2. P=[1211], D=[1002], A5=[65663334]

  3. 3. P=[1312], D=[0002], A5=[96966464]

  4. 4. P=[1312], D=[1002], A5=[94936261]

  5. 5. P=[1413], D=[1002], A5=[1251249392]

  6. 6. P=[2511], D=[1002], A5=[15631062123]

  7. 7. P=[103001010], D=[100020002], A5=[193003200032]

  8. 8. P=[011100201], D=[100010002], A5=[1623101006232]

  9. 9. P=[113001010], D=[100020002], A5=[1933103200032]

  10. 10. P=[113102020], D=[100020002], A5=[9493316261310032]

  11. 11. P=[00112423587], D=[100000001], A10=[1007852195156]

  12. 12. P=[113205050], D=[100010001], A10=[100010001]

  13. 13. P=[111011201], D=[100000001], A10=[331221001]

  14. 14. P=[112011201], D=[100000001], A10=[331221001]

  15. 15. A23A+2I=0, A3=[29282120], A4=[61604544], A1=12[2435]

  16. 16. A23A+2I=0, A3=[36701427], A4=[761503059], A1=12[31026]

  17. 17. A3+5A28A+4I=0, A3=[1210080008], A4=[145001600016], A1=12[230010001]

  18. 18. A3+4A25A+2I=0, A3=[11470100148], A4=[1301501003016], A1=12[221020021]

  19. 19. A3+5A28A+4I=0, A3=[1217080008], A4=[1451501600016], A1=12[231010001]

  20. 20. A3+5A28A+4I=0, A3=[2221714137008], A4=[4645153029150016], A1=12[131241001]

  21. 21. A3+A=0, A3=A=A=[10065221156], A4=A2=[1007852195156]. Because λ=0 is an eigenvalue, A is singular and A1 does not exist.

  22. 22. A3+A2+AI=0, A3=A=[116220114001]=A, A4=[100010001]=I, A1=A

  23. 23. A3+A=0, A3=A=[111221441], A4=A2=[331221001]. Because λ=0 is an eigenvalue, A is singular and A1 does not exist.

  24. 24. A3+A=0,  A3=A=  [553221443], A4=A2=[331221001]. Because λ=0 is an eigenvalue, A is singular and A1 does not exist.

  25. 25. xk=Akx0=[1111][1004/5]k12[1111]x0=(C0+S0)[1/21/2] as k. The long-term distribution of population is 50% city, 50% suburban.

  26. 26. xk=Akx0=[1131][1004/5]k14[1131]x0=(C0+S0)[1/43/4] as k. The long-term distribution of population is 25% city, 75% suburban.

  27. 27. xk=Akx0=[3151][1003/5]k18[1153]x0=(C0+S0)[3/85/8] as k. The long-term distribution of population is 3/8 city, 5/8 suburban.

  28. 28. xk=Akx0=[1121][1007/10]k13[1121]x0=(C0+S0)[1/32/3] as k. The long-term distribution of population is 1/3 city, 2/3 suburban.

  29. 29. xk=Akx0=[1121][10017/20]k13[1121]x0=(C0+S0)[1/32/3] as k. The long-term distribution of population is 1/3 city, 2/3 suburban.

  30. 30. xk=Akx0=[3141][10013/20]k17[1143]x0=(C0+S0)[3/74/7] as k. The long-term distribution of population is 3/7 city, 4/7 suburban.

  31. 31. xk=Akx0=[5542][1004/5]k110[2545]x0=[2.5R0F02R00.8F0] as k. The fox-rabbit population approaches a stable situation with 2.5R0F0 foxes and 2R00.8F0 rabbits.

  32. 32. xk=Akx0=[10271][19/200017/20]k14[12710]x0=[00] as k. The fox and rabbit population both die out.

  33. 33. xk=Akx0=[101093][21/20003/4]k160[310910]x0160(1.05)k(10R03F0)[109] as when k is sufficiently large. The fox and rabbit populations are both increasing at 5% per year, with 10 foxes for each 9 rabbits.

  34. 34. A=PDP1=[41305641]. If n is even, then Dn=I so An=PDnP1=PIP1=I. If n is odd, then An=An1A=IA=A. Thus A99=A and A100=I.

  35. 35. λ=±1 implies that Dn=I if n is even, in which case An=PDnP1=I.

  36. 36. A2=I, so A3=A2A=IA=A, A4=A3A=A2=I, and so forth.

  37. 37. A2=I, so A3=A2A=IA=A, A4=A3A=A2=I, and so forth.

  38. 38. If B=[0100] so B2=0, and it follows that An=I+nB=[1n01].

Chapter 7

Section 7.1

  1. 1. x1=x2, x2=7x13x2+t2

  2. 2. x1=x2, x2=x3, x3=x4,x4=x1+3x26x3+ cos 3t

  3. 3. x1=x2, t2x2=(1t2)x1tx2

  4. 4. x1=x2, x2=x3, t3x3=5x13tx2+2t2x3+ ln t

  5. 5. x1=x2, x2=x3, x3=x22+ cos x1

  6. 6. x1=x2, x2=5x14y1, y1=y2, y2=4x1+5y1

  7. 7. x1=x2, y1=y2, x2=kx1·(x12+y12)3/2, y2=ky1·(x12+y12)3/2

  8. 8. x1=x2, x2=4x1+2y13x2 y1=y2, y2=3x1y12y2+ cos t

  9. 9. x1=x2, y1=y2, z1=z2, x2=3x1y1+2z1, y2=x1+y14z1, z2=5x1y1z1

  10. 10. x1=x2, x2=x1(1y1) y1=y2, y2=y1(1x1)

  11. 11. x(t)=A cos t+B sin t, y(t)=B cos tA sin t

  12. 12. x(t)=Aet+Bet, y(t)=AetBet

  13. 13. x(t)=A cos 2t+B sin 2t, y(t)=B cos 2t+A sin 2t; x(t)=cos 2t, y(t)=sin 2t

  14. 14. x(t)=A cos 10t+B sin 10t, y(t)=B cos 10tA sin 10t; x(t)=3 cos 10t+4 sin 10t, y(t)=4 cos 10t3 sin 10t

  15. 15. x(t)=A cos 2t+B sin 2t, y(t)=4B cos 2t4A sin 2t

  16. 16. x(t)=A cos 4t+B sin 4t, y(t)=12B cos 4t12A sin 4t

  17. 17. x(t)=Ae3t+Be2t, y(t)=3Ae3t+2Be2t; x(t)=e2t, y(t)=2e2t

  18. 18. x(t)=Ae2t+Be5t, y(t)=2Ae2t+5Be5t; A=173 and B=113 in the particular solution.

  19. 19. x(t)=e2t sin 3t, y(t)=e2t(3 cos 3t+2 sin 3t)

  20. 20. x(t)=(A+Bt)e3t, y(t)=(3A+B+3Bt)e3t

Section 7.2

  1. 1. (AB)=[18t+18t21+2t12t2+32t33+3t24t38t+3t2+4t3]

  2. 3. x=[xy], P(t)=[0330], f(t)=[00]

  3. 5. x=[xy], P(t)=[2451], f(t)=[3ett2]

  4. 7. x=[xyz], P(t)=[011101110], f(t)=[000]

  5. 9. x=[xyz], P(t)=[341103067], f(t)=[tt2t3]

  6. 11. x=[x1x2x3x4], P(t)=[0100002000034000], f(t)=[0000]

  7. 13. W(t)=e3t; x(t)=[2c1et+c2e2t3c1etc2e2t]

  8. 15. W(t)4; x(t)=[c1e2t+c2e2tc1e2t+5c2e2t]

  9. 17. W(t)=7e3t; x(t)=[3c1e2t+c2e2t2c1e2t+3c2e2t]

  10. 19. W(t)3; x(t)=[c1e2t+c2etc1e2t+c3etc1e2t(c2+c3)et]

  11. 21. W(t)=e2t; x(t)=[3c1e2t+c2et+c3et2c1e2tc2etc3et2c1e2t+c2et]

  12. 23. x=2x1x2

  13. 25. x=15x14x2

  14. 27. x=x1+2x2+x3

  15. 29. x=3x13x25x3

  16. 31. x=3x1+7x2+x32x4

  17. 32. x=13x1+41x2+3x312x4

  18. 33. (a) x2=tx1, so neither is a constant multiple of the other. (b) W(x1,x2)0, whereas Theorem 2 implies that W0 if x1 and x2 were independent solutions of a system of the indicated form.

Section 7.3

  1. 1. x1(t)=c1et+c2e3t, x2(t)=c1et+c2e3t

  2. 2. x1(t)=c1et+3c2e4t, x2(t)=c1et+2c2e4t

  3. 3. General solution x1(t)=c1et+4c2e6t, x2(t)=c1et+3c2e6t Particular solution x1(t)=17(et+8e6t), x2(t)=17(et+6e6t).

  4. 4. x1(t)=c1e2t+c2e5t, x2(t)=6c1e2t+c2e5t

  5. 5. x1(t)=c1et+7c2e5t, x2(t)=c1et+c2e5t

  6. 6. General solution x1(t)=5c1e3t+c2e4t, x2(t)=6c1e3tc2e4t Particular solution x1(t)=5e3t+6e4tx2(t)=6e3t6e4t.

  7. 7. x1(t)=c1et+2c2e9t, x2(t)=c1et3c2e9t

  8. 8. x1(t)=5c1 cos 2t+5c2 sin 2t, x2(t)=(c12c2) cos 2t+(2c1+c2) sin 2t

  9. 9. General solution x1(t)=5c1 cos 4t+5c2 sin 4t, x2(t)=c1(2 cos 4t+4 sin 4t)+c2(2 sin 4t4 cos 4t). Particular solution x1(t)=2 cos 4t114sin 4t, x2(t)=3 cos 4t+12sin 4t

  10. 10. x1(t)=2c1 cos 3t2c2 sin 3t, x2(t)=(3c1+3c2) cos 3t+(3c23c1) sin 3t

  11. 11. General solution x1(t)=et(c1 cos 2tc2 sin 2t), x2(t)=et(c1 sin 2t+c2 cos 2t) Particular solution x1(t)=4et sin 2t, x2(t)=4et cos 2t

  12. 12. x1(t)=e2t(5c1 cos 2t5c2 sin 2t), x2(t)=e2t[(c1+2c2) cos 2t+(2c1+c2) sin 2t]

  13. 13. x1(t)=3e2t(c1 cos 3tc2 sin 3t), x2(t)=e2t[(c1+c2) cos 3t+(c1c2) sin 3t]

  14. 14. x1(t)=e3t(c1 cos 4t+c2 sin 4t), x2(t)=e3t(c1 sin 4tc2 cos 4t)

  15. 15. x1(t)=5e5t(c1 cos 4tc2 sin 4t), x2(t)=e5t[(2c1+4c2) cos 4t+(4c12c2) sin 4t]

  16. 16. x1(t)=c1e10t+2c2e100t, x2(t)=2c1e10t5c2e100t

  17. 17. x1(t)=c1e9t+c2e6t+c3, x2(t)=c1e9t2c2e6t, x3(t)=c1e9t+c2e6tc3

  18. 18. x1(t)=c1e9t+4c3, x2(t)=2c1e9t+c2e6tc3, x3(t)=2c1e9tc2e6tc3

  19. 19. x1(t)=c1e6t+c2e3t+c3e3t, x2(t)=c1e6t2c2e3t, x3(t)=c1e6t+c2e3tc3e3t

  20. 20. x1(t)=c1e9t+c2e6t+c3e2t, x2(t)=c1e9t2c2e6t, x3(t)=c1e9t+c2e6tc3e2t

  21. 21. x1(t)=6c1+3c2et+2c3et, x2(t)=2c1+c2et+c3et, x3(t)=5c1+2c2et+2c3et

  22. 22. x1(t)=c2et+c3e3t, x2(t)=c1e2tc2etc3e3t, x3(t)=c1e2t+c3e3t

  23. 23. x1(t)=c1e2t+c3e3t, x2(t)=c1e2t+c2e2tc3e3t, x3(t)=c2e2t+c3e3t

  24. 24. x1(t)=c1et+c2(2 cos 2t sin 2t)+c3(cos 2t+2 sin 2t) x2(t)=c1etc2(3 cos 2t+ sin 2t)+c3(cos 2t3 sin 2t) x3(t)=c2(3 cos 2t+ sin 2t)+c3(3 sin 2t cos 2t)

  25. 25. x1(t)=c1+e2t[(c2+c3) cos 3t+(c2+c3) sin 3t], x2(t)=c1+2e2t(c2 cos 3tc3 sin 3t), x3(t)=2e2t(c2 cos 3t+c3 sin 3t)

  26. 26. x1(t)=4e3tet(4cos tsin t),  x2(t)=9e3tet(9cos t+2 sin t), x3(t)=17et cos t

  27. 27. x1(t)=15e0.2t, x2(t)=15(e0.2te0.4t). The maximum amount ever in tank 2 is x2(5 ln 2)=3.75 lb.

  28. 28. x1(t)=15e0.4t, x2(t)=40(e0.4t+e0.25t). The maximum amount ever in tank 2 is about 6.85 lb.

  29. 29. x1(t)=10+5e0.6t, x2(t)=55e0.6t

  30. 30. x1(t)=15c1+c2e0.65t,x2(t)=8c1c2e0.65t.

  31. 31. x1(t)=27etx2(t)=27et27e2tx3(t)=27et54e2t+27e3t.

    The maximum amount of salt everin tank 3 is x3(ln 3)=4 pounds.

  32. 32. x1(t)=45e3t, x2(t)=135e3t+135e2t, x3(t)=135e3t270e2t+135et.

    The maximum amount of salt ever in tank 3 is x3(ln 3)=20 pounds.

  33. 33. x1(t)=45e4t, x2(t)=90e4t90e6t, x3(t)=270e4t+135e6t+135e2t.

    The maximum amount of saltever in tank 3 is x3(12 ln 3)=20 pounds.

  34. 34. x1(t)=40e3t, x2(t)=60e3t60e5t, x3(t)=150e3t+75e5t+75et.

    The maximum amount of salt ever in tank 3 is x3(12 ln 5)21.4663 pounds.

  35. 35. x1(t)=1017(55e18t216e11t), x2(t)=317(165e18t144e11t), x3(t)=20+17(220e18t360e11t). The limiting amounts of salt in tanks 1, 2, and 3 are 10 lb, 3 lb, and 20 lb.

  36. 36. x1(t)=4+e3t/5[14 cos (3t/10)2 sin (3t/10)], x2(t)=10e3t/5[10 cos (3t/10)10 sin (3t/10)], x3(t)=4e3t/5[4 cos (3t/10)+8 sin (3t/10)]. The limiting amounts of salt in tanks 1, 2, and 3 are 4 lb, 10 lb, and 4 lb.

  37. 37. x1(t)=30+e3t[25 cos (t2)+102 sin (t2)], x2(t)=10e3t[10 cos (t2)2522 sin (t2)], x3(t)=15e3t[15 cos (t2)+4522 sin (t2)]. The limiting amounts of salt in tanks 1, 2, and 3 are 30 lb, 10 lb, and 15 lb.

  38. 38. x1(t)=c1et, x2(t)=2c1et+c2e2t, x3(t)=3c1et3c2e2t+c3e3t, x4(t)=4c1et+6c2e2t4c3e3t+c4e4t

  39. 39. x1(t)=3c1et+c4e2t, x2(t)=2c1et+c3e2tc4e2t, x3(t)=4c1et+c2et, x4(t)=c1et

  40. 40. x1(t)=c1e2t, x2(t)=3c1e2t+3c2e2tc4e5t, x3(t)=c3e5t, x4(t)=c2e2t3c3e5t

  41. 41. x1(t)=2e10t+e15t=x4(t), x2(t)=e10t+2e15t=x3(t)

  42. 42. x(t)=c1[312]+c2[111]e2t+c3[231]e5t

  43. 43. x(t)=c1[315]e2t+c2[111]e4t+c3[113]e8t

  44. 44. x(t)=c1[322]e3t+c2[715]e6t+c3[533]e12t

  45. 45. x(t)=c1[1111]e3t+c2[1211]+c3[2111]e3t+c4[1121]e6t

  46. 46. x(t)=c1[3211]e4t+c2[1221]e2t+c3[1111]e4t+c4[3233]e8t

  47. 47. x(t)=c1[2211]e3t+c2[1211]e3t+c3[2111]e6t+c4[1121]e9t

  48. 48. x(t)=c1[1212]e16t+c2[2511]e32t+c3[3112]e48t+c4[1123]e64t

  49. 49. x(t)=c1[10311]e3t+c2[03011]+c3[17111]e3t+c4[01011]e6t+c5[20521]e9t

  50. 50. x(t)=c1[011101]e7t+c2[100011]e4t+c3[010101]e3t+c4[001010]e5t+c5[110001]e9t+[001110]e11t

Section 7.4

Note that phase portraits for Problems 1-16 are found in the answers for Section 7.2.

  1. 1. Saddle point (real eigenvalues of opposite sign)

  2. 2. Saddle point (real eigenvalues of opposite sign)

  3. 3. Saddle point (real eigenvalues of opposite sign)

  4. 4. Saddle point (real eigenvalues of opposite sign)

  5. 5. Saddle point (real eigenvalues of opposite sign)

  6. 6. Improper nodal source (distinct positive real eigenvalues)

  7. 7. Saddle point (real eigenvalues of opposite sign)

  8. 8. Center (pure imaginary eigenvalues)

  9. 9. Center (pure imaginary eigenvalues)

  10. 10. Center (pure imaginary eigenvalues)

  11. 11. Spiral source (complex conjugate eigenvalues with positive real part)

  12. 12. Spiral source (complex conjugate eigenvalues with positive real part)

  13. 13. Spiral source (complex conjugate eigenvalues with positive real part)

  14. 14. Spiral source (complex conjugate eigenvalues with positive real part)

  15. 15. Spiral source (complex conjugate eigenvalues with positive real part)

  16. 16. Improper nodal sink (distinct negative real eigenvalues)

  17. 17. Center; pure imaginary eigenvalues

  18. 18. Improper nodal source; distinct positive real eigenvalues; v1=[01]T, v2=[11]T

  19. 19. Saddle point; real eigenvalues of opposite sign; v1=[01]T corresponds to the negativeeigen value and v2=[11]T to the positive one.

  20. 20. Spiral source; complex conjugate eigenvalues with positive real part

  21. 21. Proper nodal source; repeated positive real eigenvalue with linearly independent eigenvectors

  22. 22. Parallel lines; one zero and one negative real eigenvalue

  23. 23. Spiral sink; complex conjugate eigenvalues with negative real part

  24. 24. Improper nodal sink; distinct negative real eigenvalues; v1=[11]T,v2=[14]T

  25. 25. Saddle point; real eigenvalues of opposite sign; v1=[11]T corresponds to the positive eigenvalue and v2=[41]T to the negative one.

  26. 26. Center; pure imaginary eigenvalues

  27. 27. Improper nodal source; distinct positive real eigenvalues; v1=[23]T, v2=[21]T.

  28. 28. Spiral sink; complex conjugate eigenvalues with negative real part

Section 7.5

  1. 1. The natural frequencies are ω0=0 and ω1=2. In the degenerate natural mode with “frequency” ω0=0 the two masses move linearly with x1(t)=x2(t)=a0+b0t. At frequency ω1=2 they oscillate in opposite directions with equal amplitudes.

  2. 2. The natural frequencies are ω1=1 and ω2=3. In the natural mode with frequency ω1, the two masses m1 and m2 move in the same direction with equal amplitudes of oscillation. At frequency ω2 they move in opposite directions with equal amplitudes.

  3. 3. The natural frequencies are ω1=1 and ω2=2. In the natural mode with frequency ω1, the two masses m1 and m2 move in the same direction with equal amplitudes of oscillation. In the natural mode with frequency ω2 they move in opposite directions with the amplitude of oscillation of m1 twice that of m2.

  4. 4. The natural frequencies are ω1=1 and ω2=5. In the natural mode with frequency ω1, the two masses m1 and m2 move in the same direction with equal amplitudes of oscillation. At frequency ω2 they move in opposite directions with equal amplitudes.

  5. 5. The natural frequencies are ω1=2 and ω2=2. In the natural mode with frequency ω1, the two masses m1 and m2 move in the same direction with equal amplitudes of oscillation. At frequency ω2 they move in opposite directions with equal amplitudes.

  6. 6. The natural frequencies are ω1=2 and ω2=8. In the natural mode with frequency ω1, the two masses m1 and m2 move in the same direction with equal amplitudes of oscillation. In the natural mode with frequency ω2 they move in opposite directions with the amplitude of oscillation of m1 twice that of m2.

  7. 7. The natural frequencies are ω1=2 and ω2=4. In the natural mode with frequency ω1, the two masses m1 and m2 move in the same direction with equal amplitudes of oscillation. At frequency ω2 they move in opposite directions with equal amplitudes.

  8. 8. x1(t)=2 cos t+3 cos 3t5 cos 5t, x2(t)=2 cos t3 cos 3t+ cos 5t. We have a superposition of three oscillations, in which the two masses move (1) in the same direction with frequency ω1=1 and equal amplitudes; (2) in opposite directions with frequency ω2=3 and equal amplitudes; (3) in opposite directions with frequency ω3=5 and with the amplitude of motion of m1 being 5 times that of m2.

  9. 9. x1(t)=5 cos t8 cos 2t+3 cos 3t, x2(t)=5 cos t+4 cos 2t9 cos 3t. We have a superposition of three oscillations, in which the two masses move (1) in the same direction with frequency ω1=1 and equal amplitudes; (2) in opposite directions with frequency ω2=2 and with the amplitude of motion of m1 being twice that of m2; (3) in opposite directions with frequency ω3=3 and with the amplitude of motion of m2 being 3 times that of m1.

  10. 10. x1(t)=15 cos 2t+ cos 4t+14 cos t, x2(t)=15 cos 2t cos 4t+16 cos t. We have a superposition of three oscillations, in which the two masses move (1) in the same direction with frequency ω1=1 and with the amplitude of motion of m2 being 8/7 times that of m1; (2) in the same direction with frequency ω2=2 and equal amplitudes; (3) in opposite directions with frequency ω3=4 and equal amplitudes.

  11. 11. (a) The natural frequencies are ω1=6 and ω2=8. In mode 1 the two masses oscillate in the same direction with frequency ω1=6 and with the amplitude of motion of m1 being twice that of m2. In mode 2 the two masses oscillate in opposite directions with frequency ω2=8 and with the amplitude of motion of m2 being 3 times that of m1. (b) x(t)=2 sin 6t+19 cos 7t, y(t)=sin 6t+3 cos 7t We have a superposition of (only two) oscillations, in which the two masses move (1) in the same direction with frequency ω1=6 and with the amplitude of motion of m1 being twice that of m2; (2) in the same direction with frequency ω3=7 and with the amplitude of motion of m1 being 19/3 times that of m2.

  12. 12. The system’s three natural modes of oscillation have (1) natural frequency ω1=2 with amplitude ratios 1:0:1; (2) natural frequency ω2=2+2 with amplitude ratios 1 : 2 : 1; (3) natural frequency ω3=22 with amplitude ratios 1 : 2 : 1.

  13. 13. The system’s three natural modes of oscillation have (1) natural frequency ω1=2 with amplitude ratios 1 : 0 : 1; (2) natural frequency ω2=4+22 with amplitude ratios 1 : 2 : 1; (3) natural frequency ω3=422 with amplitude ratios 1 : 2 : 1.

  14. 15. x1(t)=23cos 5t2 cos 53t+43cos 10t, x2(t)=43cos 5t+4 cos 53t+163cos 10t.

    We have a superposition of two oscillations with the natural frequencies ω1=5 and ω2=53 and a forced oscillation with frequency ω=10. In each of the two natural oscillations the amplitude of motion of m2 is twice that of m1, while in the forced oscillation the amplitude of motion of m2 is four times that of m1.

  15. 20. x1(t)=v0, x2(t)=0, x1(t)=v0 for t>π/2

  16. 21. x1(t)=v0, x2(t)=0, x1(t)=2v0 for t>π/2

  17. 22. x1(t)=2v0, x2(t)=v0, x1(t)=v0 for t>π/2

  18. 23. x1(t)=2v0, x2(t)=2v0, x1(t)=3v0 for t>π/2

  19. 24. (a) ω11.0293 Hz; ω21.7971 Hz. (b) v128 mi/h; v249 mi/h

  20. 27. ω1=210, v140.26 (ft/s (about 27 mi/h), ω2=55, v271.18 ft/s (about 49 mi/h)

  21. 28. ω16.1311, v139.03 ft/s (about 27 mi/h) ω210.3155, v265.67 ft/s (about 45 mi/h)

  22. 29. ω15.0424, v132.10 ft/s (about 22 mi/h), ω29.9158, v263.13 ft/s (about 43 mi/h)

Section 7.6

  1. 1. Repeated eigenvalue λ=3, eigenvector v=[11]T; x1(t)=(c1+c2+c2t)e3t, x2(t)=(c1c2t)e3t

  2. 2. Repeated eigenvalue λ=2, single eigenvector v=[11]T; x1(t)=(c1+c2+c2t)e2t, x2(t)=(c1+c2t)e2t

  3. 3. Repeated eigenvalue λ=3, eigenvector v=[22]T; x1(t)=(2c1+c22c2t)e3t, x2(t)=(2c1+2c2t)e3t

  4. 4. Repeated eigenvalue λ=4, single eigenvector v=[11]T; x1(t)=(c1+c2c2t)e4t, x2(t)=(c1+c2t)e4t

  5. 5. Repeated eigenvalue λ=5, eigenvector v=[24]T; x1(t)=(2c1+c2+2c2t)e5t, x2(t)=(4c14c2t)e5t

  6. 6. Repeated eigenvalue λ=5, single eigenvector v=[44]T; x1(t)=(4c1+c24c2t)e5t, x2(t)=(4c1+4c2t)e5t

  7. 7. Eigenvalues λ=2, 2, 9 with three linearly independent eigenvectors; x1(t)=c1e2t+c2e2t, x2(t)=c1e2t+c3e9t, x3(t)=c2e2t

  8. 8. Eigenvalues λ=7, 13, 13 with three linearly independent eigenvectors; x1(t)=2c1e7tc3e13t, x2(t)=3c1e7t+c3e13t, x3(t)=c1e7t+c2e13t

  9. 9. Eigenvalues λ=5, 5, 9 with three linearly independent eigenvectors; x1(t)=c1e5t+7c2e5t+3c3e9t, x2(t)=2c1e5t, x3(t)=2c2e5t+c3e9t

  10. 10. Eigenvalues λ=3, 3, 7 with three linearly independent eigenvectors; x1(t)=5c1e3t3c2e3t+2c3e7t, x2(t)=2c1e3t+c3e7t, x3(t)=c2e3t

  11. 11. Triple eigenvalue λ=1 of defect 2; x1(t)=(2c2+c32c3t)et, x2(t)=(c1c2+c2tc3t+12c3t2)et, x3(t)=(c2+c3t)et

  12. 12. Triple eigenvalue λ=1 of defect 2; x1(t)=et(c1+c3+c2t+12c3t2) x2(t)=et(c1+c2t+12c3t2), x3(t)=et(c2+c3t)

  13. 13. Triple eigenvalue λ=1 of defect 2; x1(t)=(c1+c2t+12c3t2)et, x2(t)=(2c2+c3+2c3t)et, x3(t)=(c2+c3t)et

  14. 14. Triple eigenvalue λ=1 of defect 2; x1(t)=et(5c1+c2+c3+5c2t+c3t+52c3t2), x2(t)=et(25c15c225c2t5c3t252c3t2), x3(t)=et(5c1+4c25c2t+4c3t52c3t2)

  15. 15. Triple eigenvalue λ=1 of defect 1; x1(t)=(3c1+c33c3t)et, x2(t)=(c1+c3t)et, x3(t)=(c2+c3t)et

  16. 16. Triple eigenvalue λ=1 of defect 1; x1(t)=et(3c1+3c2+c3) x2(t)=et(2c12c3t), x3(t)=et(2c2+2c3t)

  17. 17. Triple eigenvalue λ=1 of defect 1; x1(t)=(2c1+c2)et, x2(t)=(3c2+c3+6c3t)et, x3(t)=9(c1+c3t)et

  18. 18. Triple eigenvalue λ=1 of defect 1; x1(t)=et(c12c2+c3), x2(t)=et(c2+c3t), x3(t)=et(c12c3t)

  19. 19. Double eigenvalues λ=1 and λ=1, with four linearly independent solutions; x1(t)=c1et+c4et, x2(t)=c3et, x3(t)=c2et+3c4et, x4(t)=c1et2c3et

  20. 20. Eigenvalue λ=2 with multiplicity 4 and defect 3; x1(t)=(c1+c3+c2t+c4t+12c3t2+16c4t3)e2t, x2(t)=(c2+c3t+12c4t2)e2t, x3(t)=(c3+c4t)e2t, x4(t)=c4e2t

  21. 21. Eigenvalue λ=1 with multiplicity 4 and defect 2; x1(t)=(2c2+c32c3t)et, x2(t)=(c2+c3t)et, x3(t)=(c2+c4+c3t)et, x4(t)=(c1+c2t+12c3t2)et

  22. 22. Eigenvalue λ=1 with multiplicity 4 and defect 2; x1(t)=(c1+3c2+c4+c2t+3c3t+12c3t2)et, x2(t)=(2c2c3+2c3t)et, x3(t)=(c2+c3t)et, x4(t)=(2c1+6c2+2c2t+6c3t+c3t2)et

  23. 23. x(t)=c1v1et+(c2v2+c3v3)e3t with v1=[112]T, v2=[409]T, v3=[021]T

  24. 24. x(t)=c1v1et+(c2v2+c3v3)e3t with v1=[533]T, v2=[401]T, v3=[210]T

  25. 25. x(t)=[c1v1+c2(v1t+v2)+c3(12v1t2+v2t+v3)]e2t with v1=[101]T, v2=[410]T, and v3=[100]T

  26. 26. x(t)=[c1v1+c2(v1t+v2)+c3(12v1t2+v2t+v3)]e3t with v1=[022]T, v3=[213]T, and v3=[100]T

  27. 27. x(t)=[c1v1+c2(v1t+v2)+c3v3]e2t with v1=[538]T, v2=[100]T, v3=[100]T

  28. 28. x(t)=[c1v1+c2(v1t+v2)+c3(12v1t2+v2t+v3)]e2t with v1=[1192890]T, v2=[173417]T, and v3=[100]T

  29. 29. x(t)=[c1v1+c2(v1t+v2)]et+[c3v3+c4(v3t+v4)]e2t with v1=[1312]T, v2=[0100]T, v3=[0110]T, v4=[0021]T

  30. 30. x(t)=[c1v1+c2(v1t+v2)]et+[c3v3+c4(v3t+v4)]e2t, with v1=[0113]T, v2=[0012]T, v3=[1000]T, v4=[0035]T

  31. 31. x(t)=[c1v1+c2(v1t+v2)+c3(12v1t2+v2t+v3)+c4v4]et with v1=[4272142]T, v2=[34221027]T, v3=[1000]T, v4=[0130]T

  32. 32. x(t)=(c1v1+c2v2)e2t+(c3v3+c4v4+c5v5)e3t with v1=[80310]T, v2=[10003]T, v3=[32100]T, v4=[22030]T, v5=[11003]T

  33. 33. x1(t)=[cos 4tsin 4t00]T e3t, x2(t)=[sin 4tcos 4t00]T e3t, x3(t)=[t cos 4tt sin 4tcos 4tsin 4t]T e3t, x4(t)=[t sin 4tt cos 4tsin 4tcos 4t]T e3t

  34. 34. x1(t)=[sin 3t3 cos 3t3 cos 3t0sin 3t]e2t, x2(t)=[cos 3t3 cos 3t+3 cos 3t0cos 3t]e2t, x3(t)=[3 cos 3t+t sin 3t(3t10) cos 3t(3t+9) sin 3tsin 3tt sin 3t]e2t, x4(t)=[t cos 3t+t sin 3t(3t+9) cos 3t+(3t10) sin 3tcos 3tt cos 3t]e2t

  35. 35. x1(t)=x2(t)=v0(1et); limtx1(t)=limtx2(t)=v0

  36. 36. x1(t)=v0(22ettet), x2(t)=v0(22ettet12t2et); limt x1(t)=limt x2(t)=2v0

In Problems 37, 39, 41, 43, and 45 we give a nonsingular matrix Q and a Jordan-form matrix J such that A=QJQ1. Any scalar multiple of Q will do the same job.

  1. 37. Q=[142109290], J=[100030003]

  2. 39. Q=[141010100], J=[210021002]

  3. 41. Q=[050130181], J=[200021002]

  4. 43. Q=[1000311210102001], J=[1100010000210002]

  5. 45. Q=[3043236007722280921610803043228812], J=[1000011000110001]

Section 7.7

The format for the first eight answers is this: (x(t), y(t)) at t=0.2 by the Euler method, by the improved Euler method, by the Runge-Kutta method, and finally the actual values.

  1. 1. (0.8800, 2.5000), (0.9600, 2.6000), (1.0027, 2.6401), (1.0034, 2.6408)

  2. 2. (0.8100,0.8100), (0.8200,0.8200), (0.8187,0.8187), (0.8187,0.8187)

  3. 3. (2.8100, 2.3100), (3.2200, 2.6200), (3.6481, 2.9407), (3.6775, 2.9628)

  4. 4. (3.3100,1.6200), (3.8200,2.0400), (4.2274,2.4060), (4.2427,2.4205)

  5. 5. (0.5200, 2.9200), (0.8400, 2.4400), (0.5712, 2.4485), (0.5793, 2.4488)

  6. 6. (1.7600, 4.6800), (1.9200, 4.5600), (1.9029, 4.4995), (1.9025, 4.4999)

  7. 7. (3.1200, 1.6800), (3.2400, 1.7600), (3.2816, 1.7899), (3.2820, 1.7902)

  8. 8. (2.1600,0.6300), (2.5200,0.4600), (2.5320,0.3867), (2.5270,0.3889)

  9. 9. At t=1 we obtain (x, y)=(3.99261, 6.21770)(h=0.1) and (3.99234, 6.21768) (h=0.05); the actual value is (3.99232, 6.21768).

  10. 10. At t=1 we obtain (x, y)=(1.31498, 1.02537)(h=0.1) and (1.31501, 1.02538) (h=0.05);the actual value is (1.31501, 1.02538).

  11. 11. At t=1 we obtain (x, y)=(0.05832, 0.56664)(h=0.1) and (0.05832, 0.56665) (h=0.05);the actual value is (0.05832, 0.56665).

  12. 12. We solved x=y, y=x+ sin t, x(0)=y(0)=0. With h=0.1 and also with h=0.05 we obtain theactual value x(1.0)0.15058.

  13. 13. Runge-Kutta, h=0.1: about 1050 ft in about 7.7 s

  14. 14. Runge-Kutta, h=0.1: about 1044 ft in about 7.8 s

  15. 15. Runge-Kutta, h=1.0: about 83.83 mi in about 168 s

  16. 16. At 40°: 5.0 s, 352.9 ft; at 45°: 5.4 s, 347.2 ft; at 50°: 5.8 s, 334.2 ft (all values approximate)

  17. 17. At 39.0° the range is about 352.7 ft. At 39.5° it is 352.8; at 40°, 352.9; at 40.5°, 352.6; at 41.0°, 352.1.

  18. 18. Just under 57.5°

  19. 19. Approximately 253 ft/s

  20. 20. Maximum height: about 1005 ft, attained in about 5.6 s; range: about 1880 ft; time aloft: about 11.6 s

  21. 21. Runge-Kutta with h=0.1 yields these results: (a) 21400 ft, 46 s, 518 ft/s; (b) 8970 ft, 17.5 s;(c) 368 ft/s (at t23).

Chapter 8

Section 8.1

  1. 1. Φ(t)=[ete3tete3t], x(t)=12[5et+e3t5et+e3t]

  2. 2. Φ(t)=[1e4t22e4t], x(t)=14[3+5e4t610e4t]

  3. 3. Φ(t)=[5 cos 4t5 sin 4t2 cos 4t+4 sin 4t4 cos 4t2 sin 4t], x(t)=14[5 sin 4t4 cos 4t2 sin 4t]

  4. 4. Φ(t)=e2t[11+t1t], x(t)=e2t[1+tt]

  5. 5. Φ(t)=[2 cos 3t2 sin 3t3 cos 3t+3 sin 3t3 cos 3t+3 sin 3t], x(t)=13[3 cos 3tsin 3t3 cos 3t+6 sin 3t]

  6. 6. Φ(t)=e5t[cos 4t2 sin 4t2 cos 4t+2 sin 4t2 cos 4t2 sin 4t], x(t)=2e5t[cos 4t+sin 4tsin 4t]

  7. 7. Φ(t)=[63et2et2etet52et2et], x(t)=[12+12et+2et4+4et+et10+8et+2et]

  8. 8. Φ(t)=[0ete3te2tete3te2t0e3t], x(t)=[etet+e2te2t]

  9. 9. eAt=[2e3tet2e3t+2ete3tete3t+2et]

  10. 10. eAt=[2+3e2t33e2t2+2e2t32e2t]

  11. 11. eAt=[3e3t2e2t3e3t+3e2t2e3t2e2t2e3t+3e2t]

  12. 12. eAt=[3et+4e2t4et4e2t3et+3e2t4et3e2t]

  13. 13. eAt=[4e3t3et4e3t+4et3e3t3et3e3t+4et]

  14. 14. eAt=[8et+9e2t6et6e2t12et+12e2t9et8e2t]

  15. 15. eAt=[5e2t4et10e2t+10et2e2t2et4e2t+5et]

  16. 16. eAt=[9et+10e2t15et15e2t6et+6e2t10et9e2t]

  17. 17. eAt=12[e4t+e2te4te2te4te2te4t+e2t]

  18. 18. eAt=12[e2t+e6te2t+e6te2t+e6te2t+e6t]

  19. 19. eAt=15[4e10t+e5t2e10t2e5t2e10t2e5te10t+4e5t]

  20. 20. eAt=15[e5t+4e15t2e5t+2e15t2e5t+2e15t4e5t+e15t]

  21. 21. eAt=[1+ttt1t]

  22. 22. eAt=[1+6t4t9t16t]

  23. 23. eAt=[1+tttt2t1ttt2001]

  24. 24. eAt=[1+3t03t5t+18t217t18t23t013t]

  25. 25. eAt=[e2t5te2t0e2t], x(t)=eAt[47]

  26. 26. eAt=[e7t011te7te7t], x(t)=eAt[510]

  27. 27. eAt=[et2tet(3t+2t2)et0et2tet00et], x(t)=eAt[456]

  28. 28. eAt=[e5t0010te5te5t0(20t+150t)e5t30te5te5t], x(t)=eAt[405060]

  29. 29. eAt=[12t3t+6t24t+6t2+4t3016t3t+6t20012t0001] et, x(t)=eAt[1111]

  30. 30. eAt=e3t[10006t1009t+18t26t1012t+54t2+36t39t+18t26t1], x(t)=eAt[1111]

  31. 33. x(t)=[c1 cosh t+c2 sinh tc1 sinh t+c2 cosh t]

  32. 35. eAt=[e3t4te3t0e3t]

  33. 36. eAt=et[12t3t+4t2014t001]

  34. 37. eAt=[e2t3e2t3et13e2t(13+9t)et0et3tet00et]

  35. 38. eAt=[e5t4e10t4e5t16e10t(16+50t)e5t0e10t4e10t4e5t00e5t]

  36. 39. eAt=[et3tet12e2t(12+9t)et(51+18t)et(5136t)e2t0et3e2t3et6et(69t)e2t00e2t4e3t4e2t000e2t]

  37. 40. eAt=[e2t4te2t(4t+8t2)e2t100e3t(100+96t+32t2)e2t0e2t4te2t20e3t(20+16t)e2t00e2t4e3t4e2t000e3t]

Section 8.2

  1. 1. x(t)=73, y(t)=83

  2. 2. x(t)=18(1+12t), y(t)=14(5+4t)

  3. 3. x(t)=1756(864et+4e6t868+840t504t2), y(t)=1756(864et+3e6t+861882t+378t2)

  4. 4. x(t)=184(99e5t8e2t7et), y(t)=184(99e5t+48e2t63et)

  5. 5. x(t)=13(12et7tet), y(t)=13(67tet)

  6. 6. x(t)=1256(91+16t)et, y(t)=132(25+16t)et

  7. 7. x(t)=1410(369et+166e9t125 cos t105 sin t), y(t)=1410(369et249e9t120 cos t150 sin t)

  8. 8. x(t)=13(17 cos t+2 sin t), y(t)=13(3 cos t+5 sin t)

  9. 9. x(t)=14(sin 2t+2t cos 2t+t sin 2t), y(t)=14t sin 2t

  10. 10. x(t)=113et(4 cos t6 sin t), y(t)=113et(3 cos t+2 sin t)

  11. 11. x(t)=12(14t+e4t), y(t)=14(5+4t+e4t)

  12. 12. x(t)=t2, y(t)=t2

  13. 13. x(t)=12(1+5t)et, y(t)=52tet

  14. 14. x(t)=18(2+4te4t+2te4t), y(t)=12t(2+e4t)

  15. 15. (a) x1(t)=200(1et/10), x2(t)=400(1+et/102et/20) (b) x1(t)200 and x2(t)400 as t+ (c) Tank 1: about 6 min 56 s; tank 2: about 24 min 34 s

  16. 16. (a) x1(t)=600(1et/20), x2(t)=300(1+et/102et/20) (b) x1(t)600 and x2(t)300 as t (c) Tank 1: about 8 min 7 sec; tank 2: about 17 min 13 sec

  17. 17. x1(t)=10295et7e5t, x2(t)=9695ete5t

  18. 18. x1(t)=68110t75et+7e5t, x2(t)=7480t75et+e5t

  19. 19. x1(t)=7060t+16e3t+54e2t, x2(t)=560t32e3t+27e2t

  20. 20. x1(t)=3e2t+60te2t3e3t, x2(t)=6e2t+30te2t+6e3t

  21. 21. x1(t)=et14e2t+15e3t, x2(t)=5et10e2t+15e3t

  22. 22. x1(t)=10et7tet+10e3t5te3t, x2(t)=15et35tet+15e3t5te3t

  23. 23. x1(t)=3+11t+8t2, x2(t)=5+17t+24t2

  24. 24. x1(t)=2+t+ ln t, x2(t)=5+3t1t+3 ln t

  25. 25. x1(t)=1+8t+ cos t8 sin t, x2(t)=2+4t+2 cos t3 sin t

  26. 26. x1(t)=3 cos t32 sin t+17t cos t+4t sin t, x2(t)=5 cos t13 sin t+6t cos t+5t sin t

  27. 27. x1(t)=8t3+6t4, x2(t)=3t22t3+3t4

  28. 28. x1(t)=7+14t6t2+4t2 ln t, x2(t)=7+9t3t2+ ln t2t ln t+2t2 ln t

  29. 29. x1(t)=t cos t(ln  cos t)(sin t), x2(t)=t sin t+(ln  cos t)(cos t)

  30. 30. x1(t)=12t2 cos 2t, x2(t)=12t2 sin 2t

  31. 31. x1(t)=(9t2+4t3)et, x2(t)=6t2et, x3(t)=6tet

  32. 32. x1(t)=(44+18t)et+(44+26t)e2t, x2(t)=6et+(6+6t)e2t, x3(t)=2te2t

  33. 33. x1(t)=15t2+60t3+95t4+12t5, x2(t)=15t2+55t3+15t4, x3(t)=15t2+20t3, x4(t)=15t2

  34. 34. x1(t)=4t3+(4+16t+8t2)e2t, x2(t)=3t2+(2+4t)e2t, x3(t)=(2+4t+2t2)e2t, x4(t)=(1+t)e2t

Section 8.3

  1. 1. x(t)=12[et+e3tet+e3tet+e3tet+e3t]

  2. 2. x=15[2et+3e4t3et+3e4t2et+2e4t3et+2e4t]

  3. 3. x(t)=17[3et+4e6t4et+4e6t3et+3e6t4et+3e6t]

  4. 4. x(t)=17[e2t+6e5te2t+e5t6e2t+6e5t6e2t+e5t]

  5. 5. x(t)=16[et+7e5t7et7e5tet+e5t7ete5t]

  6. 6. x(t)=[5e3t+6e4t5e3t+5e4t6e3t6e4t6e3t5e4t]

  7. 7. x(t)=15[2e9t+3et2e9t+2et3e9t+3et3e9t+2et]

  8. 8. x(t)=12[2 cos 2t+sin 2t5 sin 2tsin 2t2 cos 2tsin 2t]

  9. 9. x(t)=14[4 cos 4t+2 sin 4t5 sin 4t4 sin 4t4 cos 4t2 sin 4t]

  10. 10. x(t)=13[3 cos 3t3 sin 3t2 sin 3t9 sin 3t3 cos 3t+3 sin 3t]

  11. 11. x(t)=et[cos 2tsin 2tsin 2tcos 2t]

  12. 12. x(t)=e2t2[2 cos 2tsin 2t5 sin 2tsin 2t2 cos 2t+sin 2t]

  13. 13. x(t)=e2t3[3 cos 3t+3 sin 3t9 sin 3t2 sin 3t3 cos 3t3 sin 3t]

  14. 14. x(t)=e3t[cos 4tsin 4tsin 4tcos 4t]

  15. 15. x(t)=e5t4[4 cos 4t+2 sin 4t5 sin 4t4 sin 4t4 cos 4t2 sin 2t]

  16. 16. x(t)=19[4e100t+5e10t2e100t+2e10t10e100t+10e10t5e100t+4e10t]

  17. 17. x(t)=16[3+e6t+2e9t2e6t+2e9t3+e6t+2e9t2e6t+2e9t4e6t+2e9t2e6t+2e9t3+e6t+2e9t2e6t+2e9t3+e6t+2e9t]

  18. 18. x(t)=118[16+e6t+2e9t4+4e9t4+4e9t4e6t+4e9t1+9e6t+8e9t19e6t+8e9t4e6t+4e9t19e6t+8e9t1+9e6t+8e9t]

  19. 19. x(t)=13[2e3t+e6te3t+e6te3t+e6te3t+e6t2e3t+e6te3t+e6te3t+e6te3t+e6t2e3t+e6t]

  20. 20. x(t)=16[3e2t+e6t+2e9t2e6t+2e9t3e2t+e6t+2e9t2e6t+e9t4e6t+2e9t2e6t+e9t3e2t+e6t+2e9t2e6t+e9t3e2t+e6t+2e9t]

  1. 21. x(t)=e3t[1+ttt1t]

  2. 22. x(t)=e2t[1+ttt1t]

  3. 23. x(t)=e3t[12t2t2t1+2t]

  4. 24. x(t)=e4t[1ttt1+t]

  5. 25. x(t)=e5t[1+2tt4t12t]

  6. 26. x(t)=e5t[14t4t4t1+4t]

  7. 27. x(t)=[e2t00e2te9te9te2t+e9t00e2t]

  8. 28. x(t)=[2e7t+3e13t2e7t+2e13t03e7t3e13t3e7t2e13t0e7t+e13te7t+e13te13t]

  9. 29. x(t)=[7e5t6e9t3e5t+3e9t21e5t+21e9t0e5t02e5t2e9te5t+e9t6e5t+7e9t]

  10. 30. x(t)=[5e3t4e7t10e3t+10e7t12e3t12e7t2e3t2e7t4e3t+5e7t6e3t6e7t00e3t]

  11. 31. eAt=14[et+5e3tete3t5et+5e3t5ete3t], x(t)=[et14e2t+15e3t5et10e2t+15e3t]

  12. 32. With eAt as in Problem 31 , x(t)=[(107t)et+(105t)e3t(1535t)et+(155t)e3t].

  13. 33. eAt=[1+3tt9t13t], x(t)=[3+11t+8t25+17t+24t2]

  14. 34. With eAt as in Problem 33 , x(t)=[2+t+ln t5+3t1t+3 ln t].

  15. 35. eAt=[cos t+ 2 sin t5 sin tsin tcos t 2 sin t], x(t)=[1+8t+cos t8 sin t2+4t+2 cos t3 sin t]

  16. 36. With eAt as in Problem 35 , x(t)=[3 cos t32 sin t+17t cos t+4t sin t5 cos t13 sin t+6t cos t+5t sin t].

  17. 37. eAt=[1+2t4tt12t], x(t)=[8t3+6t43t22t3+3t4]

  18. 38. With eAt as in Problem 37 , x(t)=[7+14t6t2+4t2 ln t7+9t3t2+ln t2t ln t+2t2 ln t].

  19. 39. eAt=[cos tsin tsin tcos t], x(t)=[t cos t(ln cos t)(sin t)t sin t+(ln cos t)(cos t)]

  20. 40. eAt=[cos 2t2 sin 2tsin 2tcos 2t], x(t)=[12t2 cos 2t12t2 sin 2t]

  21. 41. x(t)=[9et+10e3t2et+2e3t4et4e3t9et9e3t2et2e3t4et+4e3t18et+18e3t4et+4e3t8et7e3t]

  22. 42. x(t)=[5e2t+6e3t10e2t+10e3t20e2t+20e3t3e2t+3e3t6e2t+7e3t12e2t+12e3t3e2t+3e3t6e2t6e3t12e2t11e3t]

  23. 43. x(t)=12e2t[t28t+24t2+34tt2+8t2t8t+22tt24t2+2tt2+2]

  24. 44. x(t)=12e3t[4t+22t2t2t2+2tt2+2t22t26tt2+4tt24t+2]

  25. 45. x(t)=[ettettet2tet3et+(32t)e2t(13t)et(13t)ete2t2(3t1)et+(2t)e2tet+(1+2t)e2ttettet+e2t2tet+te2t2et+2e2t2tet2tet4tet+e2t]

  26. 46. x(t)=12et[48t2+68t+218t224t6t2+8t36t2+60t7t2+44t3t218t+2t2+6t6t2+38t21t220t9t2+6t3t22t+218t218t42t254t18t2+18t6t26t36t248t+2]

  27. 47. x(t)=[cos tcos t]+i[sin 3tsin 3t] There are two natural modes—one in which the two masses move in the same direction with frequency ω1=1 and with equal amplitudes, and one in which they move in opposite directions with frequency ω2=3 and with equal amplitudes.

  28. 48. x(t)=[cos tcos t]+i[sin 5 tsin 5 t] There are two natural modes—one in which the two masses move in the same direction with frequency ω1=1 and with equal amplitudes, and one in which they move in opposite directions with frequency ω2=5 and with equal amplitudes.

  29. 49. x(t)=[cos 2 tcos 2 t]+i[sin 2tsin 2t] There are two natural modes—one in which the two masses move in the same direction with frequency ω1=2 and with equal amplitudes, and one in which they move in opposite directions with frequency ω2=2 and with equal amplitudes.

  30. 50. x(t)=[cos 2tcos 2t]+i[sin 4tsin 4t] There are two natural modes-one in which the two masses move in the same direction with frequency ω1=2 and with equal amplitudes, and one in which they move in opposite directions with frequency ω2=4 and with equal amplitudes.

Chapter 9

Section 9.1

  1. 1. 6.1.14

  2. 2. 6.1.16

  3. 3. 6.1.19

  4. 4. 6.1.13

  5. 5. 6.1.12

  6. 6. 6.1.18

  7. 7. 6.1.15

  8. 8. 6.1.17

  9. 9. Equilibrium solutions x(t)0, ±2. The critical point (0, 0) in the phase plane looks like a center, whereas the points (±2, 0) look like saddle points.

  10. 10. Equilibrium solution x(t)0. The critical point (0, 0) in the phase plane looks like a spiral sink.

  11. 11. Equilibrium solutions x(t),2π,π, 0, π, 2π,. The phase portrait shown in the solutions manual suggests that the critical point (nπ, 0) in the phase plane is a spiral sink if n is even, but is a saddle point if n is odd.

  12. 12. Equilibrium solution x(t)0. The critical point (0, 0) in the phase plane looks like a spiral source, with the solution curves emanating from this source spiraling outward toward a closed curve trajectory.

  13. 13. Solution x(t)=x0e2t, y(t)=y0e2t. The origin is a stable proper node similar to the one illustrated in Fig. 6.1.4.

  14. 14. Solution x(t)=x0e2t, y(t)=y0e2t. The origin is an unstable saddle point.

  15. 15. Solution x(t)=x0e2t, y(t)=y0et. The origin is a stable node.

  16. 16. Solution x(t)=x0et, y(t)=y0e3t. The origin is an unstable improper node.

  17. 17. Solution x(t)=A cos t+B sin t, y(t)=B cos tA sin t. The origin is a stable center.

  18. 18. Solution x(t)=A cos 2t+B sin 2t, y(t)=2B cos 2t+2A sin 2t. The origin is a stable center.

  19. 19. Solution x(t)=A cos 2t+B sin 2t, y(t)=B cos 2tA sin 2t. The origin is a stable center.

  20. 20. Solution x(t)=e2t(A cos t+B sin t), y(t)=e2t[(2A+B) cos t(A+2B) sin t]. The origin is a stable spiral point.

  21. 23. The origin and the circles x2+y2=C>0; the origin is a stable center.

  22. 24. The origin and the hyperbolas y2x2=C; the origin is an unstable saddle point.

  23. 25. The origin and the ellipses x2+4y2=C>0; the origin is a stable center.

  24. 26. The origin and the ovals of the form x4+y4=C>0; the origin is a stable center.

Section 9.2

  1. 1. Asymptotically stable node

  2. 2. Unstable improper node

  3. 3. Unstable saddle point

  4. 4. Unstable saddle point

  5. 5. Asymptotically stable node

  6. 6. Unstable node

  7. 7. Unstable spiral point

  8. 8. Asymptotically stable spiral point

  9. 9. Stable, but not asymptotically stable, center

  10. 10. Stable, but not asymptotically stable, center

  11. 11. Asymptotically stable node: (2, 1)

  12. 12. Unstable improper node: (2,3)

  13. 13. Unstable saddle point: (2, 2)

  14. 14. Unstable saddle point: (3, 4)

  15. 15. Asymptotically stable spiral point: (1, 1)

  16. 16. Unstable spiral point: (3, 2)

  17. 17. Stable center: (52,12)

  18. 18. Stable, but not asymptotically stable, center: (2,1)

  19. 19. (0, 0) is a stable node. Also, there is a saddle point at (0.67, 0.40).

  20. 20. (0, 0) is an unstable node. Also, there is a saddle point at (1,1) and a spiral sink at (2.30,1.70).

  21. 21. (0, 0) is an unstable saddle point. Also, there is a spiral sink at (0.51,2.12).

  22. 22. (0, 0) is an unstable saddle point. Also, there are nodal sinks at (±0.82,±5.06) and nodal sources at (±3.65,0.59).

  23. 23. (0, 0) is a spiral sink. Also, there is a saddle point at (1.08,0.68).

  24. 24. (0, 0) is a spiral source. No other critical points are visible.

  25. 25. Theorem 2 implies only that (0, 0) is a stable sink—either a node or a spiral point. The phase portrait for 5x, y5 also shows a saddle point at (0.74,3.28) and spiral sink at (2.47,0.46). The origin looks like a nodal sink in a second phase portrait for 0.2x, y0.2, which also reveals a second saddle point at (0.12, 0.07).

  26. 26. Theorem 2 implies only that (0, 0) is an unstable source. The phase portrait for 3x, y3 also shows saddle points at (0.20, 0.25) and (0.23,1.50), as well as a nodal sink at (2.36, 0.58).

  27. 27. Theorem 2 implies only that (0, 0) is a center or a spiral point, but does not establish its stability. The phase portrait for 2x, y2 also shows saddle points at (0.25,0.51) and (1.56, 1.64), plus a nodal sink at (1.07,1.20). The origin looks like a likely center in a second phase portrait for 0.6x, y0.6.

  28. 28. Theorem 2 implies only that (0, 0) is a center or a spiral point, but does not establish its stability (though in the phase portrait it looks like a likely center). The phase portrait for 0.25x0.25, 1y1 also shows saddle points at (0.13, 0.63) and (0.12,0.47).

  29. 29. There is a saddle point at (0, 0). The other critical point (1, 1) is indeterminate, but looks like a center in the phase portrait.

  30. 30. There is a saddle point at (1, 1) and a spiral sink at (1, 1).

  31. 31. There is a saddle point at (1, 1) and a spiral sink at (1,1).

  32. 32. There is a saddle point at (2, 1) and a spiral sink at (2,1).

  33. 37. Note that the differential equation is homogeneous.

Section 9.3

  1. 1. Linearization at (0, 0): x=200x, y=150y; phase plane portrait:

  2. 3. Linearization at (75, 50): u=300v, v=100u; phase plane portrait:

  3. 5. The characteristic equation is λ2+45λ+126=0.

  4. 7. The characteristic equation is (24λ)22·(18)2=0. Phase plane portrait:

    Phase plane portrait for the nonlinear system in Problems 4-7:

  5. 9. The characteristic equation is λ2+58λ120=0.

  6. 10. The characteristic equation is (λ+36)(λ+18)576=0. Phase plane portrait:

    Phase plane portrait for the nonlinear system in Problems 8-10:

  7. 12. The characteristic equation is λ2+2λ15=0.

  8. 13. The characteristic equation is λ2+2λ+6=0. Phase plane portrait:

  9. 15. The characteristic equation is λ2+2λ24=0.

  10. 17. The characteristic equation is λ24λ+6=0. Phase plane portrait:

  11. 19. The characteristic equation is λ2+10=0. Phase plane portrait:

  12. 21. The characteristic equation is λ2λ6=0.

  13. 22. The characteristic equation is λ25λ+10=0. Phase plane portrait:

  14. 24. The characteristic equation is λ2+5λ14=0.

  15. 25. The characteristic equation is λ2+5λ+10=0. Phase plane portrait:

  16. 26. Naturally growing populations in competition Critical points: nodal source (0, 0) and saddle point (3, 2) Nonzero coexisting populations x(t)3, y(t)2

  17. 27. Naturally declining populations in cooperation Critical points: nodal sink (0, 0) and saddle point (3, 2) Nonzero coexisting populations x(t)3, y(t)2

  18. 28. Naturally declining predator, naturally growing prey population Critical points: saddle point (0, 0) and apparent stable center (4, 8)

    Nonzero coexisting populations x(t)4, y(t)8

  19. 29. Logistic and naturally growing populations in competition Critical points: nodal source (0, 0), nodal sink (3, 0), and saddle point (2, 2) Nonzero coexisting populations x(t)2, y(t)2

  20. 30. Logistic and naturally declining populations in cooperation Critical points: saddle point (0, 0), nodal sink (3, 0) and saddle point (5, 4) Nonzero coexisting populations x(t)5, y(t)4

  21. 31. Logistic prey, naturally declining predator population Critical points: saddle points (0, 0) and (3, 0), spiral sink (2, 4) Nonzero coexisting populations x(t)2, y(t)4

  22. 32. Logistic populations in cooperation

    Critical points: nodal source (0, 0), saddle points (10, 0) and (0, 20), nodal sink (30, 60)

    Nonzero coexisting populations x(t)30, y(t)60

  23. 33. Logistic prey and predator populations Critical points: nodal source (0, 0), saddle points (30, 0) and (0, 20), nodal sink (4, 22)

    Nonzero coexisting populations x(t)4, y(t)22

  24. 34. Logistic prey and predator populations Critical points: nodal source (0, 0), saddle points (15, 0) and (0, 5), spiral sink (10, 10)

    Nonzero coexisting populations x(t)10, y(t)10

Section 9.4

  1. 1. Eigenvalues: 2, 3; stable node

  2. 2. Eigenvalues: 1, 3; unstable node

  3. 3. Eigenvalues: 3, 5; unstable saddle point

  4. 4. Eigenvalues: 1±2i; stable spiral point

  5. 5. Critical points: (0, nπ) where n is an integer; an unstable saddle point if n is even, a stable spiral point if n is odd

  6. 6. Critical points: (n, 0) where n is an integer; an unstable saddle point if n is even, a stable spiral point if n is odd

  7. 7. Critical points: (nπ, nπ) where n is an integer; an unstable saddle point if n is even, a stable spiral point if n is odd

  8. 8. Critical points: (nπ, 0) where n is an integer; an unstable nodeif n is even, an unstable saddle point if n is odd

  9. 9. If n is odd then (nπ, 0) is an unstable saddle point.

  10. 10. If n is odd then (nπ, 0) is a stable node.

  11. 11. (nπ, 0) is a stable spiral point.

  12. 12. Unstable saddle points at (2, 0) and (2, 0), a stable center at (0, 0)

  13. 13. Unstable saddle points at (2, 0) and (2, 0), a stable spiral point at (0, 0)

  14. 14. Stable centers at (2, 0) and (2, 0), an unstable saddle point at (0, 0)

  15. 15. A stable center at (0, 0) and an unstable saddle point at (4, 0)

  16. 16. Stable centers at (2, 0), (0, 0) and (2, 0), unstable saddle points at (1, 0) and (1, 0)

  17. 17. (0, 0) is a spiral sink.

  18. 18. (0, 0) is a spiral sink; the points (±2, 0) are saddle points.

  19. 19. (0, 0) is a spiral sink.

  20. 20. (nπ, 0) is a spiral sink if n is even, a saddle point if n is odd.

Chapter 10

Section 10.1

  1. 1. 1/s2, s>0

  2. 2. 2/s3, s>0

  3. 3. e/(s3), s>3

  4. 4. s/(s2+1), s>0

  5. 5. 1/(s21), s>1

  6. 6. 12[1/ss/(s2+4)], s>0

  7. 7. (1es)/s, s>0

  8. 8. (ese2s)/s, s>0

  9. 9. (1esses)/s2, s>0

  10. 10. (s1+es)/s2, s>0

  11. 11. 12πs3/2+3s2, s>0

  12. 12. (45π192s3/2)/(8s7/2), s>0

  13. 13. s22(s3)1, s>3

  14. 14. 3π/(4s5/2)+1/(s+10), s>0

  15. 15. s1+s(s225)1, s>5

  16. 16. (s+2)/(s2+4), s>0

  17. 17. cos2 2t=12(1+ cos 4t); 12[s1+s/(s2+16)], s>0

  18. 18. 3/(s2+36), s>0

  19. 19. s1+3s2+6s3+6s4, s>0

  20. 20. 1/(s1)2, s>1

  21. 21. (s24)/(s2+4)2, s>0

  22. 22. 12[s/(s236)s1]

  23. 23. 12t3

  24. 24. 2t/π

  25. 25. 183t3/2π1/2

  26. 26. e5t

  27. 27. 3e4t

  28. 28. 3 cos 2t+12sin 2t

  29. 29. 53sin 3t3 cos 3t

  30. 30. cosh 2t92 sinh 2t

  31. 31. 35 sinh 5t10 cosh 5t

  32. 32. 2u(t3)

  33. 37. f(t)=1u(ta). Your figure should indicate that the graph of f contains the point (a, 0), but not the point (a, 1).

  34. 38. f(t)=u(ta)u(tb). Your figure should indicate that the graph of f contains the points (a, 1) and (b, 0), but not the points (a, 0) and (b, 1).

  35. 39. Figure 10.2.8 shows the graph of the unit staircase function.

Section 10.2

  1. 1. x(t)=5 cos 2t

  2. 2. x(t)=3 cos 3t+43sin 3t

  3. 3. x(t)=23(e2tet)

  4. 4. x(t)=12(7e3t3e5t)

  5. 5. x(t)=13(2 sin tsin 2t)

  6. 6. x(t)=13(cos t cos 2t)

  7. 7. x(t)=18(9 cos tcos 3t)

  8. 8. x(t)=19(1 cos 3t)

  9. 9. x(t)=16(23et+e3t)

  10. 10. x(t)=14(2t3+12et9e2t)

  11. 11. x(t)=1, y(t)=2

  12. 12. x(t)=29(e2tet3tet), y(t)=19(e2tet+6tet)

  13. 13. x(t)=(2/3) sinh (t/3), y(t)=cosh (t/3)+(1/3) sinh (t/3)

  14. 14. x(t)=14(2t3 sin 2t), y(t)=18(2t+3 sin 2t)

  15. 15. x(t)=13(2+e3t/2 [cos (rt/2)+r sin (rt/2)]), y(t)=121(289et+2e3t/2 [cos (rt/2)+4r sin (rt/2)]) where r=3

  16. 16. x(t)=cos t+ sin t, y(t)=et cos t, z(t)=2 sin t

  17. 17. f(t)=13(e3t1)

  18. 18. f(t)=35(1e5t)

  19. 19. f(t)=14(1 cos 2t)=12sin2 t

  20. 20. f(t)=19(6 sin 3t cos 3t+1)

  21. 21. f(t)=t sin t

  22. 22. f(t)=19(1+ cosh 3t)

  23. 23. f(t)=t+ sinh t

  24. 24. f(t)=12(e2t2et+1)

Section 10.3

  1. 1. 24/(sπ)5

  2. 2. 34π(s+4)5/2

  3. 3. 3π/[(s+2)2+9π2]

  4. 4. 2(2s+5)/(4s2+4s+17)

  5. 5. 32e2t

  6. 6. (tt2)et

  7. 7. te2t

  8. 8. e2t cos t

  9. 9. e3t(3 cos 4t +72sin 4t)

  10. 10. 136e2t/3(8 cos 43t5 sin 43t)

  11. 11. 12 sinh 2t

  12. 12. 2+3e3t

  13. 13. 3e2t5e5t

  14. 14. 2+e2t3et

  15. 15. 125(e5t15t)

  16. 16. 1125[e2t(5t2)+e3t(5t+2)]

  17. 17. 116(sinh 2tsin 2t)

  18. 18. e4t(1+12t+24t2+323t3)

  19. 19. 13(2 cos 2t+2 sin 2t2 cos t sin t)

  20. 20. 132[e2t(2t1)+e2t(2t+1)]

  21. 21. 12et(5 sin t3t cos t2t sin t)

  22. 22. 164et/2[(4t+8) cos t+(43t) sin t]

  23. 27. 14e3t(8 cos 4t+9 sin 4t)

  24. 28. 14(12e2t+e4t)

  25. 29. 18(6t+3 sinh 2t)

  26. 30. 110[2ete2t(2 cos 2t+ sin 2t)]

  27. 31. 115(6e2t5e3t)

  28. 32. 12(cosh t+ cos t)

  29. 33. x(t)=r(cosh rt sin rt sinh rt cos rt) where r=1/2

  30. 34. 12sin 2t+13sin 3t

  31. 35. 116(sin 2t2t cos 2t)

  32. 36. 150[2e2t+(10t2) cos t(5t+14) sin t]

  33. 37. 150[(5t1)et+e2t(cos 3t+32 sin 3t)]

  34. 38. 1510e3t(489 cos 3t+307 sin 3t)+ 1170(7 cos 2t+6 sin 2t)

  35. 39.

Section 10.4

  1. 1. 12t2

  2. 2. (eatat1)/a2

  3. 3. 12(sin tt cos t)

  4. 4. 2(t sin t)

  5. 5. teat

  6. 6. (eatebt)/(ab)

  7. 7. 13(e3t1)

  8. 8. 14(1 cos 2t)

  9. 9. 154(sin 3t3t cos 3t)

  10. 10. (kt sin kt)/k3

  11. 11. 14(sin 2t+2t cos 2t)

  12. 12. 15[1e2t(cos t+2 sin t)]

  13. 13. 110(3e3t3 cos t+sin t)

  14. 14. 13(cos t cos 2t)

  15. 15. 6s/(s2+9)2, s>0

  16. 16. (2s324s)/(s2+4)3, s>0

  17. 17. (s24s5)/(s24s+13)2, s>0

  18. 18. 2(3s2+6s+7)(s+1)2(s2+2s+5)2, s>0

  19. 19. 12πarctans=arctan(1/s), s>0

  20. 20. 12ln(s2+4)ln s, s>0

  21. 21. ln sln(s3), s>3

  22. 22. ln (s+1) ln (s1), s>1

  23. 23. (2 sinh 2t)/t

  24. 24. 2(cos 2t cos t)/t

  25. 25. e2t+e3t2 cos t)/t

  26. 26. (e2t sin 3t)/t

  27. 27. 2(1cos t)/t

  28. 28. 18(t sin tt2 cos t)

  29. 29. (s+1)X(s)+4X(s)=0; x(t)=Ct3et, C0

  30. 30. X(s)=A/(s+3)3; x(t)=Ct2e3t, C0

  31. 31. (s2)X(s)+3X(s)=0; x(t)=Ct2e2t, C0

  32. 32. (s2+2s)X(s)+(4s+4)X(s)=0; x(t)=C(1te2tte2t), C0

  33. 33. (s2+1)X(s)+4sX(s)=0; x(t)=C(sin tt cos t), C0

  34. 34. x(t)=Ce2t(sin 3t3t cos 3t), C0

Section 10.5

  1. 1. f(t)=u(t3)·(t3)

  2. 2. f(t)=(t1)u(t1)(t3)u(t3)

  3. 3. f(t)=u(t1)·e2(t1)

  4. 4. f(t)=et1u(t1)e2et2u(t2)

  5. 5. f(t)=u(tπ)· sin (tπ)=u(tπ) sin t

  6. 6. f(t)=u(t1)· cos π(t1)=u(t1) cos πt

  7. 7. f(t)=sin tu(t2π) sin (t2π)=[1u(t2π)] sin t

  8. 8. f(t)=cos πtu(t2) cos π(t2)=[1u(t2)] cos πt

  9. 9. f(t)=cos πt+u(t3) cos π(t3)=[1u(t3)] cos πt

  10. 10. f(t)=2u(tπ) cos 2(tπ)2u(t2π) cos 2(t2π)=2[u(tπ)u(t2π)] cos 2t

  11. 11. f(t)=2[1u3(t)]; F(s)=2(1e3s)/s

  12. 12. F(s)=(ese4s)/s

  13. 13. F(s)=(1e2πs)/(s2+1)

  14. 14. F(s)=s(1e2s)/(s2+π2)

  15. 15. F(s)=(1+e3πs)/(s2+1)

  16. 16. F(s)=2(eπse2πs)/(s2+4)

  17. 17. F(s)=π(e2s+e3s)/(s2+π2)

  18. 18. F(s)=2π(e3s+e5s)/(4s2+π2)

  19. 19. F(s)=es(s1+s2)

  20. 20. F(s)=(1es)/s2

  21. 21. F(s)=(12es+e2s)/s2

  22. 28. F(s)=(1easaseas)/[s2(1e2as)]

  23. 31. x(t)=12[1u(tπ)] sin2 t

  24. 32. x(t)=g(t)u(t2)g(t2), where g(t)=112(34et+e4t)

  25. 33. x(t)=18[1u(t2π)](sin t13sin 3t)

  26. 34. x(t)=g(t)u(t1)[g(t1)+h(t1)], where g(t)=t sin t and h(t)=1 cos t

  27. 35. x(t)=14{1+t+(t+1)e2t+ u(t2)[1t+(3t5)e2(t2)]}

  28. 36. x(t)=2| sin t| sin t

  29. 37. x(t)=g(t)+2n=1(1)nu(tnπ)g(tnπ), where g(t)=113et(3 cos 3t+ sin 3t)

Chapter 11

Section 11.1

  1. 1. y(x)=c0(1+x+x22+x33!+)=c0ex; ρ=+

  2. 2. y(x)=c0(1+4x1!+42x22!+43x33!+44x44!+)=c0e4x; ρ=

  3. 3. y(x)=c0(13x2+(3x)22!22(3x)33!23+(3x)44!24)=c0e3x/2; ρ=+

  4. 4. y(x)=c0(1x21!+x42!x63!+)=c0ex2; ρ=

  5. 5. y(x)=c0(1+x33+x62!32+x93!33+)=c0exp(13x3); ρ=+

  6. 6. y(x)=c0(1+x2+x24+x38+x416+)=2c02x; ρ=2

  7. 7. y(x)=c0(1+2x+4x2+8x3+)=c012x; ρ=12

  8. 8. y(x)=c0(1+x2x28+x3165x4128+)=c01+x; ρ=1

  9. 9. y(x)=c0(1+2x+3x2+4x3+ )=c0(1x)2; ρ=1

  10. 10. y(x)=c0(13x2+3x28+x316+3x4128+) =c0(1x)3/2; ρ=1

  11. 11. y(x)=c0(1+x22!+x44!+x66!+)+ c1(x+x33!+x55!+x77!+) =c0 cosh x+c1 sinh x; ρ=+

  12. 12. y(x)=c0(1+(2x)22!+(2x)44!+(2x)66!+)+ c12((2x)+(2x)33!+(2x)55!+(2x)77!+) =c0 cosh 2x+c12 sinh 2x; ρ=

  13. 13. y(x)=c0(1(3x)22!+(3x)44!(3x)66!+)+ c13(3x(3x)33!+(3x)55!(3x)77!+) =c0 cos 3x+13c1 sin 3x; ρ=+

  14. 14. y(x)=x+c0(1x22!+x44!x66!+) +(c11)(xx33!+x55!x77!+) =x+c0 cos x+(c11) sin x; ρ=

  15. 15. (n+1)cn=0 for all n0, so cn=0 for all n0.

  16. 16. 2ncn=cn for all n0, so cn=0 for all n0.

  17. 17. c0=c1=0 and cn+1=ncn for n1, thus cn=0 for all n0.

  18. 18. cn=0 for all n0

  19. 19. (n+1)(n+2)cn+2=4cn; y(x)=32[(2x)(2x)33!+(2x)55!(2x)77!+]=32sin 2x

  20. 20. (n+1)(n+2)cn+2=4cn; y(x)=2[1+(2x)22!+(2x)44!+(2x)66!+]=2 cosh 2x

  21. 21. n(n+1)cn+1=2ncncn1; y(x)=x+x2+x32!+x43!+x54!+=xex

  22. 22. n(n+1)cn+1=ncn+2cn1; y=e2x

  23. 23. As c0=c1=0 and (n2n+1)cn+(n1)cn1=0 for n2, cn=0 for all n0

Section 11.2

  1. 1. cn+2=cn; y(x)=c0n=0x2n+c1n=0x2n+1=c0+c1x1x2; ρ=1

  2. 2. cn+2=12cn; ρ=2; y(x)=c0n=0(1)nx2n2n+c1n=0(1)nx2n+12n

  3. 3. (n+2)cn+2=cn; y(x)=c0n=0(1)nx2nn!2n+c1n=0(1)nx2n+1(2n+1)!!; ρ=+

  4. 4. (n+2)cn+2=(n+4)cn; ρ=1; c0n=0(1)n(n+1)x2n+13c1n=0(1)n(2n+3)x2n+1

  5. 5. 3(n+2)cn+2=ncn; ρ=3; y(x)=c0+c1n=0x2n+1(2n+1)3n

  6. 6. (n+1)(n+2)cn+2=(n3)(n4)cn; ρ=; y(x)=c0(1+6x2+x4)+c1(x+x3)

  7. 7. 3(n+1)(n+2)cn+2=(n4)2cn; y(x)=c0(18x23+8x427)+c1(xx32+x5120+9n=3(1)n[(2n5)!!]2x2n+1(2n+1)!3n)

  8. 8. 2(n+1)(n+2)cn+2=(n4)(n+4)cn; y(x)=c0(14x2+2x4)+c1(x5x34+7x532+n=3(2n5)!!(2n+3)!!x2n+1(2n+1)!2n)

  9. 9. (n+1)(n+2)cn+2=(n+3)(n+4)cn; ρ=1; y(x)=c0n=0(n+1)(2n+1)x2n+c13n=0(n+1)(2n+3)x2n+1

  10. 10. 3(n+1)(n+2)cn+2=(n4)cn; y(x)=c0(1+2x23+x427)+c1(x+x36+x5360+3n=3(1)n(2n5)!!x2n+1(2n+1)!3n)

  11. 11. 5(n+1)(n+2)cn+2=2(n5)cn; y(x)=c1(x4x315+4x5375)+c0(1x2+x410+x6750+15n=4(2n7)!!2nx2n(2n)!5n)

  12. 12. c2=0; (n+2)cn+3=cn; y(x)=c0(1+n=1x3n2·5(3n1))+c1n=0x3n+1n!3n

  13. 13. c2=0; (n+3)cn+3=cn; y(x)=c0n=0(1)nx3nn!3n+c1n=0(1)nx3n+11·4(3n+1)

  14. 14. c2=0; (n+2)(n+3)cn+3=cn; y(x)=c0(1+n=1(1)nx3n3n·n!·2·5(3n1))+c1n=0(1)nx3n+13n·n!·1·4(3n+1)

  15. 15. c2=c3=0; (n+3)(n+4)cn+4=cn; y(x)=c0(1+n=1(1)nx4n4n·n!·3·7(4n1))+c1n=0(1)nx4n+14n·n!·5·9(4n+1)

  16. 16. y(x)=x

  17. 17. y(x)=1+x2

  18. 18. y(x)=2n=0(1)n(x1)2nn!2n; converges for all x

  19. 19. y(x)=13n=0(2n+3)(x1)2n+1; converges if 0<x<2

  20. 20. y(x)=26(x3)2; converges for all x

  21. 21. y(x)=1+4(x+2)2; converges for all x

  22. 22. y(x)=2x+6

  23. 23. 2c2+c0=0; (n+1)(n+2)cn+2+cn+cn1=0 for n1; y1(x)=1x22x36+; y2(x)=xx36x412+

  24. 24. y1(x)=1+x33+x55+x645+; y2(x)=x+x33+x46+x55+

  25. 25. c2=c3=0, (n+3)(n+4)cn+4+(n+1)cn+1+cn=0 for n0; y1(x)=1x412+x7126+; y2(x)=xx412x520+

  26. 26. y(x)=c0(1x630+x972+)+c1(xx742+x1090+)

  27. 27. y(x)=1xx22+x33x424+x530+29x6720+13x7630143x840320+; y(0.5)0.4156

  28. 28. y(x)=c0(1x22+x36+)+c1(xx36+x412+)

  29. 29. y1(x)=112x2+1720x6+; y2(x)=x16x3160x5+

  30. 30. y(x)=c0(1x22+x36+)+c1(xx22+x418+)

  31. 33. The following figure shows the interlaced zeros of the 4th and 5th Hermite polynomials.

  32. 34. The figure below results when we use n=40 terms in each summation. But with n=50 we get the same picture as Fig. 8.2.3 in the text.

Section 11.3

  1. 1. Ordinary point

  2. 2. Ordinary point

  3. 3. Irregular singular point

  4. 4. Irregular singular point

  5. 5. Regular singular point; r1=0, r2=1

  6. 6. Regular singular point; r1=1, r2=2

  7. 7. Regular singular point; r=3, 3

  8. 8. Regular singular point; r=12, 3

  9. 9. Regular singular point x=1

  10. 10. Regular singular point x=1

  11. 11. Regular singular points x=1, 1

  12. 12. Irregular singular point x=2

  13. 13. Regular singular points x=2, 2

  14. 14. Irregular singular points x=3, 3

  15. 15. Regular singular point x=2

  16. 16. Irregular singular point x=0, regular singular point x=1

  17. 17. y1(x)=cos x, y2(x)=sin x

  18. 18. y1(x)=n=0xnn!(2n+1)!!, y2(x)=x1/2n=0xnn!(2n1)!!

  19. 19. y1(x)=x3/2(1+3n=1xnn!(2n+3)!!), y2(x)=1xn=2xnn!(2n3)!!

  20. 20. y1(x)=x1/3n=0(1)n2nxnn!·4·7(3n+1), y2(x)=n=0(1)n2nxnn!25(3n1)

  21. 21. y1(x)=x(1+n=1x2nn!·7·11(4n+3)), y2(x)=x1/2(1+n=1x2nn!15(4n3))

  22. 22. y1(x)=x3/2(1+n=1(1)nx2nn!·9·13(4n+5)), y2(x)=x1(1+n=1(1)n1x2nn!37(4n1))

  23. 23. y1(x)=x1/2(1+n=1x2n2n·n!·19·31(12n+7)), y2(x)=x2/3(1+n=1x2n2nn!517(12n7))

  24. 24. y1(x)=x1/3(1+n=1(1)nx2n2n·n!·7·13(6n+1)), y2(x)=1+n=1(1)nx2n2nn!511(6n+1)

  25. 25. y1(x)=x1/2n=0(1)nxnn!·2n=x1/2ex/2, y2(x)=1+n=1(1)nxn(2n1)!!

  26. 26. y1(x)=x1/2n=0x2nn!·2n=x1/2exp(12x2), y2(x)=1+n=12nx2n37(4n1)

  27. 27. y1(x)=1xcos 3x, y2(x)=1xsin 3x

  28. 28. y1(x)=1x cosh 2x, y2(x)=1x sinh 2x

  29. 29. y1(x)=1xcos x2, y2(x)=1xsin x2

  30. 30. y1(x)=cos x2, y2(x)=sin x2

  31. 31. y1(x)=x1/2 cosh x, y2(x)=x1/2 sinh x

  32. 32. y1(x)=x+x25, y2(x)=x1/2(15x215x285x348+)

  33. 33. y1(x)=x1(1+10x+5x2+10x39+), y2(x)=x1/2(1+11x2011x2224+671x324192+)

  34. 34. y1(x)=x(1x242+x41320+), y2(x)=x1/2(17x224+19x43200+)

Section 11.4

  1. 5. J4(x)=1x2(x224)J0(x)+8x3(6x2)J1(x)

  2. 10. 3

  3. 11. x2J1(x)+xJ0(x)J0(x) dx+C

  4. 12. (x34x)J1(x)+2x2J0(x)+C

  5. 13. (x49x2)J1(x)+(3x39x)J0(x)+9J0(x) dx+C

  6. 14. xJ1(x)+J0(x) dx+C

  7. 15. 2xJ1(x)x2J0(x)+C

  8. 16. 3x2J1(x)+(3xx3)J0(x)3J0(x) dx+C

  9. 17. (4x316x)J1(x)+(8x2x4)J0(x)+C

  10. 18. 2J1(x)+J0(x) dx+C

  11. 19. y(x)=x[c1J0(x)+C2Y0(x)]

  12. 20. y(x)=1x[c1J1(x)+c2Y1(x)]

  13. 21. y(x)=x[c1J1/2(3x2)+c2J1/2(3x2)]

  14. 22. y(x)=x3[c1J2(2x1/2)+c2Y2(2x1/2)]

  15. 23. y(x)=x1/3[c1J1/3(13x3/2)+c2J1/3(13x3/2)]

  16. 24. y(x)=x1/4[c1J0(2x3/2)+c2Y0(2x3/2)]

  17. 25. y(x)=x1[c1J0(x)+c2Y0(x)]

  18. 26. y(x)=x2[c1J1(4x1/2)+c2Y1(4x1/2)]

  19. 27. y(x)=x1/2[c1J1/2(2x3/2)+c2J1/2(2x3/2)]

  20. 28. y(x)=x1/4[c1J3/2(25x5/2)+c2J3/2(25x5/2)]

  21. 29. y(x)=x1/2[c1J1/6(13x3)+c2J1/6(13x3)]

  22. 30. y(x)=x1/2[c1J1/5(45x5/2)+c2J1/5(45x5/2)]